An adjunction was defined as a natural bijection between two families of morphisms: the maps
\(F(A) \to B\) on one side, the maps \(A \to G(B)\) on the other. That definition is the most
symmetric, but it is rarely the most convenient. Two further descriptions of the same relation are
in everyday use, and this page develops both. The first repackages the entire bijection into a
single pair of natural transformations. The second recasts it as a universal property, the form in
which adjunctions are most often met in practice. The three descriptions are equivalent, and
proving so is the central business of what follows.
We begin with the observation that an adjunction carries, hidden inside it, two distinguished
families of morphisms. Suppose \(F : \mathscr{A} \to \mathscr{B}\) is left adjoint to
\(G : \mathscr{B} \to \mathscr{A}\), so that for each \(A\) and \(B\) we have the bijection
\[
\mathscr{B}\bigl(F(A), B\bigr) \cong \mathscr{A}\bigl(A, G(B)\bigr),
\]
written \(g \mapsto \bar{g}\).
The bijection can be applied to any morphism, and there is one morphism always available to feed
it: the identity. Taking \(B = F(A)\), the identity \(1_{F(A)} : F(A) \to F(A)\) lives in the
left-hand set, and its transpose
\[
\eta_A := \overline{1_{F(A)}} : A \longrightarrow GF(A)
\]
lives in the right-hand set. Dually, taking \(A = G(B)\), the identity
\(1_{G(B)} : G(B) \to G(B)\) lives in the right-hand set, and its inverse transpose
\[
\varepsilon_B := \overline{1_{G(B)}} : FG(B) \longrightarrow B
\]
lives in the left-hand set. We have begun to suppress brackets, writing \(GF(A)\) for \(G(F(A))\)
and \(FG(B)\) for \(F(G(B))\). For each object we obtain one such morphism, and the naturality of
the adjunction makes these families compatible with all morphisms in sight.
Definition: Unit and Counit of an Adjunction
Let \(F \dashv G\) be an adjunction between \(F : \mathscr{A} \to \mathscr{B}\)
and \(G : \mathscr{B} \to \mathscr{A}\). The
natural transformations
\[
\eta : 1_{\mathscr{A}} \longrightarrow G \circ F, \quad
\varepsilon : F \circ G \longrightarrow 1_{\mathscr{B}},
\]
with components \(\eta_A = \overline{1_{F(A)}} : A \to GF(A)\) and
\(\varepsilon_B = \overline{1_{G(B)}} : FG(B) \to B\), are called the unit
and the counit of the adjunction, respectively.
The families \(\eta\) and \(\varepsilon\) are genuine natural transformations, not merely families
of morphisms indexed by objects. Their naturality is a consequence of the two naturality equations
of the adjunction bijection. For the unit, let \(p : A' \to A\) and transpose the morphism
\(F(p) : F(A') \to F(A)\) in two ways. Writing \(F(p) = F(p) \circ 1_{F(A')}\) and using the first
naturality equation gives \(\overline{F(p)} = GF(p) \circ \eta_{A'}\). Writing
\(F(p) = 1_{F(A)} \circ F(p)\) and using the second naturality equation, in the form
\(\overline{g \circ F(p)} = \bar{g} \circ p\) obtained by transposing both of its sides, gives
\(\overline{F(p)} = \eta_A \circ p\). Hence \(GF(p) \circ \eta_{A'} = \eta_A \circ p\), which is
the naturality square for \(\eta\). The dual computation, with the roles of the two equations
exchanged, gives the naturality of \(\varepsilon\).
The unit goes from the identity functor on \(\mathscr{A}\) to the round trip \(GF\), and the
counit from the round trip \(FG\) back to the identity functor on \(\mathscr{B}\). They are the
adjunction's measure of how far the two functors are from being mutually inverse. If \(F\) and
\(G\) were actually inverse, \(GF\) and \(FG\) would be the identity functors, and \(\eta\) and
\(\varepsilon\) would be identity natural transformations.
The Free Vector Space, Revisited
The free-forgetful adjunction \(F \dashv U\) between \(\mathbf{Set}\) and \(\mathbf{Vect}_k\)
makes the unit and counit concrete. Recall that \(F\) sends a set \(S\) to the
vector space
\(F(S)\) with basis \(S\), and \(U\) sends a vector space to its underlying set. The
adjunction is the
free-forgetful bijection
\(\mathbf{Vect}_k(F(S), V) \cong \mathbf{Set}(S, U(V))\), which restricts a linear map to the
basis and extends a function off it.
The component of the unit at a set \(S\) is the transpose of the identity
linear map
\(1_{F(S)}\). Restricting that identity to the basis gives the function
\[
\eta_S : S \longrightarrow UF(S), \quad s \longmapsto s,
\]
the inclusion of the set \(S\) as the basis of the free space \(F(S)\). Each generator is regarded
as an element of the underlying set of formal linear combinations.
The component of the counit at a vector space \(V\) is the inverse transpose of the identity
function \(1_{U(V)}\). Extending that function linearly off the basis gives the linear map
\[
\varepsilon_V : FU(V) \longrightarrow V, \quad \textstyle\sum_{v \in V} \lambda_v\, v
\longmapsto \sum_{v \in V} \lambda_v\, v,
\]
which sends a formal linear combination of vectors of \(V\) to its actual value
in \(V\). The space \(FU(V)\) is vast. It has one basis vector for every element of \(V\), so for
\(k = \mathbb{R}\) and \(V = \mathbb{R}^2\) it has uncountably many basis vectors. The map
\(\varepsilon_V\) collapses this enormous free space onto the two-dimensional \(V\) by actually
performing the additions and scalings that were, in \(FU(V)\), merely recorded as formal symbols.
The unit records the generators, and the counit evaluates them.
The Triangle Identities
The unit and counit are not independent of one another. They are bound by two equations, and these
equations turn out to carry the entire force of the adjunction. That fact takes some work to
establish and is the subject of the next section. Here we state the equations and prove they hold.
To form the equations we must compose unit and counit components with the functors applied to
them. Recall that a functor carries morphisms to morphisms, so \(F\) may be applied to a component
\(\eta_A : A \to GF(A)\) of the unit to yield a morphism \(F(\eta_A) : F(A) \to FGF(A)\) in
\(\mathscr{B}\). Likewise, \(\varepsilon\) has a component
\(\varepsilon_{F(A)} : FGF(A) \to F(A)\) at the object \(F(A)\). These two are composable, and
their composite is a morphism \(F(A) \to F(A)\). The first triangle identity asserts that this
composite is the identity.
Symmetrically, \(G(\varepsilon_B) : GFG(B) \to G(B)\) and \(\eta_{G(B)} : G(B) \to GFG(B)\)
compose to a morphism \(G(B) \to G(B)\), and the second triangle identity asserts that this too
is the identity.
Lemma: The Triangle Identities
Let \(F \dashv G\) be an adjunction with unit \(\eta\) and counit \(\varepsilon\).
Then for every object \(A\) of \(\mathscr{A}\) and every object \(B\) of
\(\mathscr{B}\), the identities
\[
\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)}, \quad
G(\varepsilon_B) \circ \eta_{G(B)} = 1_{G(B)}
\]
hold.
The triangle identities. Each triangle commutes. Going up and back down the diagonal returns
the identity. The left triangle lives in \(\mathscr{B}\), the right in \(\mathscr{A}\).
Proof:
We prove the first identity. The second follows by the dual argument, which exchanges the
roles of \(F\) and \(G\) and of unit and counit. The proof turns on two facts about the
adjunction bijection \(g \mapsto \bar{g}\). First, transposing twice returns the original
morphism, \(\bar{\bar{g}} = g\). Second, it satisfies the second naturality equation, which
says that precomposing a morphism on the \(\mathscr{A}\)-side by some \(p\) corresponds, after
transposing, to precomposing the transpose by \(F(p)\) on the \(\mathscr{B}\)-side:
\[
\overline{f \circ p} = \bar{f} \circ F(p).
\]
Recall first that the counit component \(\varepsilon_{F(A)} : FGF(A) \to F(A)\) is, by
definition, the inverse transpose of the identity \(1_{GF(A)}\). Equivalently, under the
bijection \(\mathscr{B}(FGF(A), F(A)) \cong \mathscr{A}(GF(A), GF(A))\), it satisfies
\(\overline{\varepsilon_{F(A)}} = 1_{GF(A)}\). We write this as
\(\varepsilon_{F(A)} = \bar{f}\) with \(f = 1_{GF(A)}\).
Now read the naturality equation \(\overline{f \circ p} = \bar{f} \circ F(p)\) with
\(f = 1_{GF(A)}\) and \(p = \eta_A : A \to GF(A)\). Its right-hand side is
\(\bar{f} \circ F(p) = \varepsilon_{F(A)} \circ F(\eta_A)\), and its left-hand side is the
transpose of \(f \circ p = 1_{GF(A)} \circ \eta_A = \eta_A\). Hence
\[
\overline{\eta_A} = \varepsilon_{F(A)} \circ F(\eta_A).
\]
But \(\eta_A = \overline{1_{F(A)}}\) by definition, and transposing twice returns the original
morphism, so \(\overline{\eta_A} = 1_{F(A)}\). Therefore
\[
\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)},
\]
which is the first triangle identity. The second is obtained by the dual
computation in \(\mathscr{A}\).
The identities are called triangular because each is a commuting triangle, as
drawn above: one leg built from the unit, one from the counit, and the third side the identity.
A compact notation is widespread. Write \(F\eta\) for the family of morphisms \(F(\eta_A)\) and
\(\varepsilon F\) for the family \(\varepsilon_{F(A)}\). The first applies the functor \(F\) to
the components of \(\eta\), and the second evaluates \(\varepsilon\) at the objects \(F(A)\). In
this notation the two identities read
\[
(\varepsilon F) \circ (F\eta) = 1_F, \quad (G\varepsilon) \circ (\eta G) = 1_G,
\]
equalities of transformations between functors. We will use the longer componentwise form, which
requires no notation beyond the composition of morphisms already in hand. The abbreviated form is
recorded only because it is the one most often seen.
Units and Counits Determine the Adjunction
We have extracted a unit and a counit from an adjunction and shown they satisfy the triangle
identities. The central fact is that nothing is lost in the extraction. The unit and counit, two
natural transformations subject only to the triangle identities, contain the whole adjunction.
From them the entire bijection can be reconstructed, and the reconstruction is forced. An
adjunction and a triangle-compatible pair \((\eta, \varepsilon)\) are therefore two presentations
of one and the same structure.
The reconstruction rests on a formula expressing each transpose through the unit, or each inverse
transpose through the counit.
Lemma: Transpose Formulae
Let \(F \dashv G\) be an adjunction with unit \(\eta\) and counit \(\varepsilon\). Then the
transpose of any morphism is computed by
\[
\bar{g} = G(g) \circ \eta_A \quad \text{for } g : F(A) \to B,
\]
and its inverse by
\[
\bar{f} = \varepsilon_B \circ F(f) \quad \text{for } f : A \to G(B).
\]
Proof:
For the first formula, take any \(g : F(A) \to B\) and write it as
\(g = g \circ 1_{F(A)}\). The first naturality equation of the adjunction states that
postcomposing a morphism on the \(\mathscr{B}\)-side, then transposing, agrees with
transposing and then applying \(G\): \(\overline{q \circ h} = G(q) \circ \bar{h}\). Reading
it with \(h = 1_{F(A)}\) and \(q = g\) gives
\[
\bar{g} = \overline{g \circ 1_{F(A)}} = G(g) \circ \overline{1_{F(A)}} = G(g) \circ \eta_A,
\]
since \(\overline{1_{F(A)}} = \eta_A\) by definition of the unit. The second formula follows
in the same way from the second naturality equation
\(\overline{f \circ p} = \bar{f} \circ F(p)\), read with \(1_{G(B)}\) in place of \(f\) and
the given \(f : A \to G(B)\) in place of \(p\), together with the definition
\(\overline{1_{G(B)}} = \varepsilon_B\) of the counit.
The triangle identities are precisely what makes the two formulae inverse to one another, and the
next theorem carries out that computation in full.
With the formulae in hand we can state the equivalence precisely. Recall that an
adjunction
between \(F\) and \(G\) is a choice, for each \(A\) and \(B\), of the natural bijection between
\(\mathscr{B}(F(A), B)\) and \(\mathscr{A}(A, G(B))\).
Theorem: The Unit-Counit Characterization
Let \(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\) be functors.
There is a one-to-one correspondence between the following two kinds of data:
Adjunctions between \(F\) and \(G\), with \(F\) on the left and \(G\) on the right.
Pairs \((\eta, \varepsilon)\) of natural transformations
\(\eta : 1_{\mathscr{A}} \to G \circ F\) and
\(\varepsilon : F \circ G \to 1_{\mathscr{B}}\) satisfying the triangle
identities.
Under the correspondence, the pair attached to an adjunction is its unit and counit, and the
adjunction attached to a pair has \(\eta\) and \(\varepsilon\) as its unit and counit.
Proof:
We have already produced, from any adjunction, a pair \((\eta, \varepsilon)\) satisfying
the triangle identities, namely the unit and counit of the preceding sections. It
remains to show this passage is a bijection: that every triangle-compatible pair arises
from a unique adjunction.
From a pair to an adjunction. Suppose given natural transformations
\(\eta : 1_{\mathscr{A}} \to GF\) and \(\varepsilon : FG \to 1_{\mathscr{B}}\) satisfying the
triangle identities. For each \(A\) and \(B\) define functions in both directions between
\(\mathscr{B}(F(A), B)\) and \(\mathscr{A}(A, G(B))\), both denoted by a bar, by the formulae
the lemma forces them to obey:
\[
\begin{align*}
\bar{g} &= G(g) \circ \eta_A \quad (g : F(A) \to B), \\\\
\bar{f} &= \varepsilon_B \circ F(f) \quad (f : A \to G(B)).
\end{align*}
\]
We claim these are mutually inverse. Given \(g : F(A) \to B\),
\[
\begin{align*}
\bar{\bar{g}} &= \varepsilon_B \circ F(\bar{g}) \\\\
&= \varepsilon_B \circ FG(g) \circ F(\eta_A) \\\\
&= g \circ \varepsilon_{F(A)} \circ F(\eta_A) \\\\
&= g \circ 1_{F(A)} \\\\
&= g,
\end{align*}
\]
where the third equality is the naturality of \(\varepsilon\) (applied to \(g : F(A) \to B\),
giving \(\varepsilon_B \circ FG(g) = g \circ \varepsilon_{F(A)}\)) and the fourth is the first
triangle identity. Dually, for \(f : A \to G(B)\) the naturality of \(\eta\) and the second
triangle identity give \(\bar{\bar{f}} = f\). Hence the two functions are inverse bijections.
These bijections are natural in \(A\) and \(B\). Naturality reduces to the functoriality of
\(F\) and \(G\) together with the naturality of \(\eta\) and \(\varepsilon\), a direct check
on the defining formulae. The family of bijections is therefore an adjunction. Finally, its
unit and counit are the transformations we started with. The unit component at \(A\) is the
transpose of \(1_{F(A)}\), which by the first formula is
\(G(1_{F(A)}) \circ \eta_A = 1_{GF(A)} \circ \eta_A = \eta_A\), and dually the counit
component is \(\varepsilon_B\). So the adjunction we built has \((\eta, \varepsilon)\) as
its unit-counit pair.
Uniqueness. Any adjunction with unit \(\eta\) and counit \(\varepsilon\) must
compute its transposes by the formulae of the lemma, since those were derived from the unit
and counit alone. Two adjunctions with the same unit and counit therefore have the same
transpose operation on every morphism, hence are the same adjunction. The passage from pairs
to adjunctions is thus inverse to the passage from adjunctions to pairs, and the two are in
bijection.
Adjunction as a quadruple
The theorem licenses a change of viewpoint that organizes the entire subject. An adjunction
may be regarded as a quadruple \((F, G, \eta, \varepsilon)\): two functors and two natural
transformations satisfying the triangle identities. This is the form in which adjunctions
propagate to higher structures.
Every such quadruple induces, on the composite endofunctor \(T = G \circ F\), the structure of
a monad. The unit \(\eta\) serves as the monad's unit, and the counit
supplies a multiplication \(\mu = G \varepsilon F : T^2 \to T\). The unit laws of the triple
\((T, \eta, \mu)\) are the two triangle identities read on \(T\), and its associativity is the
naturality of \(\varepsilon\). The adjunction is not itself a monad. It is the richer datum
from which the monad is extracted, and this passage is one of the tools on which the
categorical account of learning systems later draws. That setting is one where the unit-counit
form is the usable one. The hom-set bijection does not transport to structures without
hom-sets, whereas a pair of transformations satisfying triangle-like equations does.
Universal Properties as Initial Objects
A third description of adjointness remains, the one met most often in ordinary mathematics. It
recasts the adjunction as a universal property. A left adjoint exists exactly when a certain
object, characterized by a mapping-out property, can be found for each input. The same form
was already glimpsed when free objects were described as initial among structured objects
receiving a map. We now name the category in which that initiality lives, and prove the
characterization in full.
The free vector space states its universal property in the familiar idiom. Given a vector space
\(V\), every function \(f : S \to U(V)\) from the generating set extends uniquely to a linear map
\(\bar{f} : F(S) \to V\), and the extension is recorded by the diagram
\[
\begin{array}{ccc}
S & \xrightarrow{\eta_S} & UF(S) \\
& {\scriptstyle f}\searrow & \downarrow{\scriptstyle U(\bar{f})} \\
& & U(V)
\end{array}
\]
commuting, with \(\eta_S\) the unit inclusion of the basis. The pattern of a universal map
\(\eta_S\) through which all others factor is initiality in disguise, and the disguise is lifted
by assembling the relevant maps into a category.
The Comma Category
Fix a functor \(G : \mathscr{B} \to \mathscr{A}\) and an object \(A\) of \(\mathscr{A}\). The maps
out of \(A\) into the \(G\)-image of some object form the objects of a category, and the maps in
\(\mathscr{B}\) compatible with them form its morphisms.
Definition: Comma Category \((A \Rightarrow G)\)
Let \(G : \mathscr{B} \to \mathscr{A}\) be a functor and \(A\) an object of \(\mathscr{A}\).
The comma category \((A \Rightarrow G)\) has:
as objects, pairs \((B, f)\) with \(B\) an object of \(\mathscr{B}\) and
\(f : A \to G(B)\) a morphism of \(\mathscr{A}\).
as morphisms \((B, f) \to (B', f')\), the morphisms
\(q : B \to B'\) of \(\mathscr{B}\) for which \(G(q) \circ f = f'\), that is, for which
the triangle
\[
\begin{array}{ccc}
A & \xrightarrow{f} & G(B) \\
& {\scriptstyle f'}\searrow & \downarrow{\scriptstyle G(q)} \\
& & G(B')
\end{array}
\]
commutes.
Composition and identities are inherited from \(\mathscr{B}\).
The usual way of naming objects of \((A \Rightarrow G)\) is a mild abuse. An object is properly
the pair \((B, f)\), but one speaks casually of "the object \(f : A \to G(B)\)," even though
distinct objects \(B, B'\) may share the same \(G\)-image.
The construction specializes in two directions worth naming. Taking \(G\) to be the identity
functor on a category \(\mathscr{A}\) and writing the object as \(A\), the comma category
\((A \Rightarrow 1_{\mathscr{A}})\) has as objects the morphisms out of \(A\). This comma category
is the coslice category under \(A\). Dually, the maps into a fixed
object \(A\) form the slice category over \(A\), whose objects are pairs
\((X, h)\) with \(h : X \to A\) and whose morphisms are commuting triangles. Slice and coslice
categories are the comma construction in its simplest guise, and they recur throughout geometry
and logic wherever objects are studied relative to a fixed base.
The Unit as an Initial Object
With the comma category named, the universal property of the free object acquires its exact
statement. The diagram for the free vector space is the assertion that the unit
\(\eta_S : S \to UF(S)\), regarded as an object \((F(S), \eta_S)\) of \((S \Rightarrow U)\), is
initial.
That is, every object \((V, f)\) receives exactly one morphism from it. The same initiality holds
for every adjunction.
Lemma: The Unit Component is Initial
Let \(F \dashv G\) be an adjunction with unit \(\eta\), and let \(A\) be an object of
\(\mathscr{A}\). Then the unit component \(\eta_A : A \to GF(A)\), regarded as the object
\((F(A), \eta_A)\) of the comma category \((A \Rightarrow G)\), is an initial object.
Proof:
Let \((B, f : A \to G(B))\) be an arbitrary object of \((A \Rightarrow G)\). We must show
there is exactly one morphism \((F(A), \eta_A) \to (B, f)\). By definition such a morphism is
a map \(q : F(A) \to B\) in \(\mathscr{B}\) making the triangle
\[
\begin{array}{ccc}
A & \xrightarrow{\eta_A} & GF(A) \\
& {\scriptstyle f}\searrow & \downarrow{\scriptstyle G(q)} \\
& & G(B)
\end{array}
\]
commute, that is, a \(q\) with \(G(q) \circ \eta_A = f\). But the transpose formula computes
\(G(q) \circ \eta_A = \bar{q}\) for every \(q : F(A) \to B\). The commuting condition is
therefore \(\bar{q} = f\). Because transposing is a bijection, this holds for exactly one
\(q\), namely \(q = \bar{f}\), the inverse transpose of \(f\). Hence there is a unique
morphism from \((F(A), \eta_A)\) to \((B, f)\), and \((F(A), \eta_A)\) is initial.
The lemma is the result foreshadowed when left adjoints were first discussed. What was promised
there only descriptively is now exact. There, a left adjoint was said to assign to each \(A\) an
initial object in the category of maps out of \(A\) into a \(G\)-image, with \(F(A)\) the object
whose \(G\)-image receives that initial morphism. Here the associated category is
\((A \Rightarrow G)\), and the initiality is the lemma above. The uniqueness of initial objects
then re-proves that a left adjoint is determined up to canonical isomorphism.
The Third Characterization
The lemma turns into a characterization of adjointness, completing the trio of equivalent
definitions. An adjunction is the same data as a unit whose every component is initial.
Theorem: The Initial-Object Characterization
Let \(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\) be functors.
There is a one-to-one correspondence between the following two kinds of data:
Adjunctions between \(F\) and \(G\), with \(F\) on the left and \(G\) on the right.
Natural transformations \(\eta : 1_{\mathscr{A}} \to G \circ F\) such that, for
every object \(A\), the component \(\eta_A : A \to GF(A)\) is an initial object of
\((A \Rightarrow G)\).
Proof:
An adjunction yields such an \(\eta\). Its unit is natural, and each component is initial by
the lemma. Conversely, suppose \(\eta : 1_{\mathscr{A}} \to GF\) is natural with every
\(\eta_A\) initial in \((A \Rightarrow G)\). We recover the adjunction through the unit-counit
characterization, by producing a counit \(\varepsilon\) for which \((\eta, \varepsilon)\)
satisfies the triangle identities. That characterization then supplies a unique adjunction
with \(\eta\) as unit.
Define \(\varepsilon_B : FG(B) \to B\), for each object \(B\), as the unique morphism
\((FG(B), \eta_{G(B)}) \to (B, 1_{G(B)})\) in the comma category
\((G(B) \Rightarrow G)\), which exists and is unique because \(\eta_{G(B)}\) is
initial there. Unwinding the comma-category condition, \(\varepsilon_B\) is the unique
map with \(G(\varepsilon_B) \circ \eta_{G(B)} = 1_{G(B)}\). This equation is already
the second triangle identity.
The family \((\varepsilon_B)_B\) is natural by an initiality argument. For \(q : B \to B'\),
both \(q \circ \varepsilon_B\) and \(\varepsilon_{B'} \circ FG(q)\) are morphisms
\(FG(B) \to B'\), and each is a morphism \((FG(B), \eta_{G(B)}) \to (B', G(q))\) in
\((G(B) \Rightarrow G)\). For the first this is the defining equation of \(\varepsilon_B\),
and for the second it is the defining equation of \(\varepsilon_{B'}\) together with the
naturality of \(\eta\) at \(G(q)\):
\[
\begin{align*}
G(q \circ \varepsilon_B) \circ \eta_{G(B)} &= G(q) \circ G(\varepsilon_B) \circ \eta_{G(B)} \\\\
&= G(q), \\\\
G(\varepsilon_{B'} \circ FG(q)) \circ \eta_{G(B)} &= G(\varepsilon_{B'}) \circ \eta_{G(B')} \circ G(q) \\\\
&= G(q).
\end{align*}
\]
Since \(\eta_{G(B)}\) is initial, there is exactly one such morphism, so the two coincide.
This is the naturality square for \(\varepsilon\).
The first triangle identity, \(\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)}\), holds by a
second initiality argument, this time in \((A \Rightarrow G)\). Both
\(\varepsilon_{F(A)} \circ F(\eta_A)\) and \(1_{F(A)}\) are endomorphisms of \(F(A)\). We
check that each is a morphism \((F(A), \eta_A) \to (F(A), \eta_A)\) in the comma category,
that is, satisfies \(G(-) \circ \eta_A = \eta_A\). For \(1_{F(A)}\) this is immediate. For
the composite, we use that \(G\) preserves composition, then the naturality of \(\eta\) at
the morphism \(\eta_A\), which gives
\(GF(\eta_A) \circ \eta_A = \eta_{GF(A)} \circ \eta_A\), and finally the defining property
\(G(\varepsilon_{F(A)}) \circ \eta_{GF(A)} = 1_{GF(A)}\) of \(\varepsilon_{F(A)}\):
\[
\begin{align*}
G(\varepsilon_{F(A)} \circ F(\eta_A)) \circ \eta_A
&= G(\varepsilon_{F(A)}) \circ GF(\eta_A) \circ \eta_A \\\\
&= G(\varepsilon_{F(A)}) \circ \eta_{GF(A)} \circ \eta_A \\\\
&= \eta_A.
\end{align*}
\]
Both endomorphisms thus name morphisms \((F(A), \eta_A) \to (F(A), \eta_A)\). Since \(\eta_A\)
is initial, there is exactly one such morphism, so the two coincide.
With both triangle identities in force, the unit-counit characterization furnishes an
adjunction having \(\eta\) and \(\varepsilon\) as its unit and counit, and \(\eta\) as its
unit in particular. This adjunction is the only one with unit \(\eta\). Any adjunction with
unit \(\eta\) has a counit satisfying the second triangle identity, which forces that counit
to be the \(\varepsilon\) just constructed, and the unit-counit characterization then allows
only one adjunction with this pair.
The three faces of one relation
Adjointness has now been described three ways. It is a natural bijection of hom-sets, a pair
of natural transformations satisfying the triangle identities, and a universal arrow given by
a unit whose components are initial objects. The first is the most symmetric, exhibiting no
preference between \(F\) and \(G\). The second is the most portable, surviving the passage to
higher structures where hom-sets are unavailable. The third is the most practical, the form in
which a working mathematician usually recognizes an adjunction in the wild, through the
observation that an object has the universal property that every map factors through it
uniquely. Because all three describe the identical structure, a fact proved in any one idiom
is available in the others, and this is what gives the theory of adjoints its reach.
When a Left Adjoint Exists
The initial-object characterization answers a question the hom-set definition leaves awkward to
pose. Given a functor \(G : \mathscr{B} \to \mathscr{A}\), does a left adjoint exist at all? The
hom-set definition presupposes one already has both functors in hand. It tests a relation
between \(F\) and \(G\) and cannot be consulted before \(F\) is produced. The initial-object
form refers to \(G\) alone. It converts the existence of a left adjoint into the existence of
initial objects in a family of categories built from \(G\), a condition one can examine without
guessing \(F\) in advance.
Corollary: Existence of a Left Adjoint
A functor \(G : \mathscr{B} \to \mathscr{A}\) has a left adjoint if and only if, for every
object \(A\) of \(\mathscr{A}\), the comma category \((A \Rightarrow G)\) has an initial
object.
Proof:
If \(G\) has a left adjoint \(F\), then its unit makes each \(\eta_A\) an initial object of
\((A \Rightarrow G)\), so initial objects exist. This is the lemma already proved.
Conversely, suppose every \((A \Rightarrow G)\) has an initial object. For each \(A\), choose
one and write it as \((F(A), \eta_A : A \to GF(A))\). Here the symbols \(F(A)\) and \(\eta_A\)
are merely names for the chosen object and its structure map.
This assignment extends to a functor. Given \(f : A \to A'\), the composite
\(\eta_{A'} \circ f : A \to GF(A')\) is an object of \((A \Rightarrow G)\), and initiality of
\(\eta_A\) supplies a unique morphism from \(\eta_A\) to it, namely a unique
\(F(f) : F(A) \to F(A')\) with \(GF(f) \circ \eta_A = \eta_{A'} \circ f\). Uniqueness forces
\(F\) to respect identities and composites, so \(F\) is a functor. The same equation says
exactly that \(\eta\) is a natural transformation \(1_{\mathscr{A}} \to GF\). Each component
\(\eta_A\) is initial by construction, so by the initial-object characterization \(F\) is left
adjoint to \(G\).
The reading from right to left is the one that does work. It manufactures the left adjoint,
object by object, out of nothing but a supply of initial objects. The functoriality of \(F\)
and the naturality of \(\eta\) are not assumed but extracted, each forced by the uniqueness
clause in the definition of initiality. A universal property, imposed at every object
compatibly, is a functor. This mechanism lies behind the recurring slogan that universal
constructions are automatically functorial.
Toward the adjoint functor theorems
The corollary is the gateway to a deeper question. It reduces the existence of a left adjoint
to the existence of initial objects, but offers no guarantee that those initial objects can be
found. The problem is shifted, not settled. The general theory supplies conditions under which
the required initial objects are guaranteed to exist, conditions phrased in terms of
completeness of \(\mathscr{B}\), preservation of limits by \(G\), and a smallness restriction
that prevents the relevant constructions from outgrowing the available sets. Those results,
the adjoint functor theorems,
take this corollary as their starting point. To build a left adjoint, one builds initial
objects, and the theorems say when the building can be done. We do not pursue them here, but
the path from this page leads directly to them.
The initial-object characterization also settles, retroactively, the status of the universal
properties met earlier across the curriculum. Free objects, quotients, and tensor products were
each pinned down by a mapping-out property and proved unique in its own setting, and products by
the dual mapping-in property. Each is now seen as the initial object of its own comma category,
or, for products, as the terminal object of the dual construction, a single phenomenon wearing
many costumes. The theory of adjoints is, in this sense, the theory of those universal properties
all at once.