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
\[
\eta : 1_{\mathscr{A}} \Longrightarrow G \circ F, \qquad
\varepsilon : F \circ G \Longrightarrow 1_{\mathscr{B}},
\]
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.
That \(\eta\) and \(\varepsilon\) are genuine natural transformations — not merely families of
morphisms indexed by objects — is a consequence of the naturality of the adjunction bijection,
and we take it as part of the definition; it is verified directly from the two naturality
equations established for the bijection. 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: were \(F\) and \(G\) actually inverse, \(GF\) and \(FG\) would be
the identity functors and \(\eta\), \(\varepsilon\) 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
\[
\eta_S : S \longrightarrow UF(S), \qquad s \longmapsto s,
\]
the inclusion of the set \(S\) as the basis of the free space \(F(S)\) — each generator,
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
\[
\varepsilon_V : FU(V) \longrightarrow V, \qquad \textstyle\sum_{v \in V} \lambda_v\, v
\longmapsto \sum_{v \in V} \lambda_v\, v,
\]
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
\(V = \mathbb{R}^2\) it is a vector space with uncountably many basis vectors, and
\(\varepsilon_V\) collapses this enormous free space down 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; 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 — a fact that 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
composites
\[
\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)}, \qquad
G(\varepsilon_B) \circ \eta_{G(B)} = 1_{G(B)}
\]
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, exchanging the roles of
\(F\) and \(G\) and of unit and counit. The proof turns on two facts about the adjunction
bijection \(g \mapsto \bar{g}\): that it is a bijection, so two morphisms sharing a transpose
are equal; and its second naturality equation, 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 transpose the composite \(\varepsilon_{F(A)} \circ F(\eta_A)\). Reading the naturality
equation \(\overline{f \circ p} = \bar{f} \circ F(p)\) with \(\bar{f} = \varepsilon_{F(A)}\),
\(f = 1_{GF(A)}\), and \(p = \eta_A\), its right-hand side \(\bar{f} \circ F(p)\) is precisely
our composite, so its transpose is the left-hand side:
\[
\overline{\varepsilon_{F(A)} \circ F(\eta_A)} = \overline{f \circ p} = 1_{GF(A)} \circ \eta_A
= \eta_A.
\]
But \(\eta_A = \overline{1_{F(A)}}\) by definition, so \(\varepsilon_{F(A)} \circ F(\eta_A)\)
and \(1_{F(A)}\) have the same transpose. The transpose is a bijection, hence injective, so the
two are equal:
\[
\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)},
\]
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. Writing \(F\eta\) for the family of morphisms
\(F(\eta_A)\) and \(\varepsilon F\) for the family \(\varepsilon_{F(A)}\) — each a transformation
built by applying a functor to the components of a natural transformation — the two identities
read
\[
(\varepsilon F) \circ (F\eta) = 1_F, \qquad (G\varepsilon) \circ (\eta G) = 1_G,
\]
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 remarkable fact — the main content of this section — 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 is the
dual statement, obtained from the second naturality equation and the definition
\(\overline{1_{G(B)}} = \varepsilon_B\) of the counit, with \(f = 1_{G(B)} \circ f\) and the
bar read in the reverse direction.
The two formulae are inverse to one another, and the triangle identities are precisely what makes
them so: composing them and simplifying by the naturality of unit and counit leaves a triangle
composite that the identities collapse to the identity morphism. The bijectivity of the transpose
is thus not separate from the triangle identities but a consequence of them — a point the next
theorem turns into the main structural result, and whose computation we carry out there 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
adjunctions between \(F\) and \(G\) (with \(F\) on the left and \(G\) on the right); and
pairs \((\eta, \varepsilon)\) of natural transformations
\(\eta : 1_{\mathscr{A}} \Rightarrow G \circ F\) and
\(\varepsilon : F \circ G \Rightarrow 1_{\mathscr{B}}\) 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 — 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
\(\eta : 1_{\mathscr{A}} \Rightarrow GF\) and \(\varepsilon : FG \Rightarrow 1_{\mathscr{B}}\)
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:
\[
\bar{g} = G(g) \circ \eta_A \quad (g : F(A) \to B), \qquad
\bar{f} = \varepsilon_B \circ F(f) \quad (f : A \to G(B)).
\]
We claim these are mutually inverse. Given \(g : F(A) \to B\),
\[
\bar{\bar{g}} = \varepsilon_B \circ F(\bar{g}) = \varepsilon_B \circ FG(g) \circ F(\eta_A)
= g \circ \varepsilon_{F(A)} \circ F(\eta_A) = g \circ 1_{F(A)} = g,
\]
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 \(\mu = G \varepsilon F : T^2
\Rightarrow T\), the triple \((T, \eta, \mu)\) inheriting its coherence from the triangle
identities. The adjunction is not itself a monad — it is the richer datum from which the monad
is extracted — but the passage from one to the other is the mechanism through which the
categorical account of learning systems is later assembled. That development requires the
unit–counit form specifically: the hom-set bijection does not transport to those settings,
whereas a pair of transformations satisfying triangle-like equations does.
Universal Properties as Initial Objects
A third description of adjointness remains, and it is 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. This is
the form 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 — 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 notation \((A \Rightarrow G)\) 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 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:
every object \((V, f)\) receives exactly one morphism from it. This 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 \(\bar{q} = f\), which holds for exactly one \(q\) — namely \(q = \bar{f}\), the
inverse transpose of \(f\) — because transposing is a bijection. Hence there is a unique
morphism from \((F(A), \eta_A)\) to \((B, f)\), and \((F(A), \eta_A)\) is initial.
This is the result foreshadowed when left adjoints were first discussed. What was then promised
only descriptively — that a left adjoint assigns to each \(A\) an initial object in the category of
maps out of \(A\) into a \(G\)-image, with \(F(A)\) the source of that initial morphism — is now
exact: 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
adjunctions between \(F\) and \(G\) (with \(F\) on the left and \(G\) on the right); and
natural transformations \(\eta : 1_{\mathscr{A}} \Rightarrow G \circ F\) 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 \(\eta : 1_{\mathscr{A}} \Rightarrow GF\) 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 \(G(\varepsilon_B) \circ \eta_{G(B)} =
1_{G(B)}\) — which is already the second triangle identity. That the family
\((\varepsilon_B)_B\) is natural follows by a now-familiar initiality argument: for
\(q : B \to B'\), both \(q \circ \varepsilon_B\) and \(\varepsilon_{B'} \circ FG(q)\) are
morphisms with the same image under the initiality of \(\eta_{G(B)}\), hence equal. 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 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, apply
\(G\) and precompose with \(\eta_A\). Since \(G\) preserves composition,
\(G(\varepsilon_{F(A)} \circ F(\eta_A)) = G(\varepsilon_{F(A)}) \circ GF(\eta_A)\), so we must
evaluate \(G(\varepsilon_{F(A)}) \circ GF(\eta_A) \circ \eta_A\). The naturality of \(\eta\) at
\(\eta_A\) gives \(GF(\eta_A) \circ \eta_A = \eta_{GF(A)} \circ \eta_A\), and the defining
property \(G(\varepsilon_{F(A)}) \circ \eta_{GF(A)} = 1_{GF(A)}\) of \(\varepsilon_{F(A)}\)
then yields \(G(\varepsilon_{F(A)} \circ F(\eta_A)) \circ \eta_A = \eta_A\). Both endomorphisms
thus name
morphisms \((F(A), \eta_A) \to (F(A), \eta_A)\); since \(\eta_A\) is initial there is exactly
one, so the two coincide. With both triangle identities in force, the unit–counit
characterization furnishes a unique adjunction having \(\eta\) and \(\varepsilon\) as its unit
and counit, and \(\eta\) as its unit in particular. Uniqueness of the adjunction follows from
uniqueness in that characterization, since \(\varepsilon\) was forced at every step.
The three faces of one relation
Adjointness has now been described three ways: as a natural bijection of hom-sets; as a pair
of natural transformations satisfying the triangle identities; and as a universal arrow, 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 — "this object has
the universal property that every map factors through it uniquely." That all three describe
the identical structure is what gives the theory of adjoints its reach: a fact proved in any
one idiom is available in the others.
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 \((F(A), \eta_A : A \to GF(A))\) — the symbols \(F(A)\) and \(\eta_A\)
being 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
\(1_{\mathscr{A}} \Rightarrow GF\). 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 is the mechanism 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 — that 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 \(\mathscr{B}\) 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, products, and tensor
products were each pinned down by a mapping property and proved unique in its own setting; each is
now seen as the initial object of its own comma category, a single phenomenon wearing many
costumes. The theory of adjoints is, in this sense, the theory of those universal properties all
at once.