Adjunctions from Universal Arrows

Everyday mathematics states adjointness neither as a hom-set bijection nor as a unit–counit pair, but as an extension property: given a vector space $V$ , any function $f : S \to V$ extends uniquely to a linear map $\overset{ˉ}{f} : F (S) \to V$ . A family of such statements, one per object, is a third complete description of an adjunction, and the most useful for constructing adjoints, since it builds a left adjoint one object at a time, with functoriality following automatically.¹

The comma category of an object over a functor

$f$ extends uniquely along $η_{S}$ is a statement about a category whose objects are arrows. The general gadget was introduced with the constructions on categories; we need one special case.

An object of $(A \Rightarrow G)$ is a way of mapping $A$ into the image of $G$ , and a map between two such is a map downstairs in $B$ that is compatible upstairs. Slice and coslice categories are the special cases where $G$ is an identity functor.²

An object of the comma category

(A \Rightarrow G)

is an arrow

A \to G (B)

; a map to another object

(B^{'}, f^{'})

is a map

q

in the target category making the triangle commute.

Following Leinster's convention, we casually call $f : A \to G (B)$ an object of $(A \Rightarrow G)$ , leaving the pair $(B, f)$ implicit.²

The unit is initial

Recall that an initial object of a category is one with exactly one map to every object; initial objects are unique up to unique isomorphism. The next lemma ties the unit of an adjunction to initiality.

The unit component

u_{A} = η_{A}

is initial in

(A \Rightarrow G)

: every arrow

f

from

A

into the image of

G

factors through it by a unique map, namely the transpose of

f

For the free vector space, $(S \Rightarrow U)$ has as objects the functions from $S$ into underlying sets of vector spaces, and initiality of $η_{S} : S \to U F (S)$ says: every function $f : S \to U (V)$ factors as $U (\overset{ˉ}{f}) \circ η_{S}$ for a unique linear $\overset{ˉ}{f} : F (S) \to V$ . Extends uniquely = is initial.¹

Barr & Wells take this property as their definition of adjointness: $F$ is left adjoint to $G$ when there is a natural transformation $η : 1 \Rightarrow GF$ such that every $f : A \to G (B)$ equals $G (q) \circ η_{A}$ for a unique $q : F (A) \to B$ — the universal mapping property of $η$ , also called the map-lifting property.⁵ In the language of universal elements, $η_{A}$ is a universal element of the functor $A (A, G (-)) : B \to Set$ .⁶

The third formulation

The lemma has a converse, giving the full three-way equivalence.

Initiality converts every verification into two maps out of an initial object agree — no computation, only uniqueness. This is the standard idiom for working with universal properties, first met in the uniqueness-up-to-isomorphism lemma.

The three equivalent presentations of an adjunction and what converts each into the others: transposing identities, recovering the bijection, and reading off initiality.

Formulation	What must be checked	Typical use
hom-set bijection	a bijection for each pair, natural both ways	recognizing an adjunction already in hand
unit and counit	two natural transformations, two identities	formal arguments; composing adjunctions
universal arrows	one initial object per object of $A$	constructing an adjoint that does not yet exist

Building a left adjoint objectwise

The universal-arrow formulation has a corollary that the other two do not make visible: to produce a left adjoint, no functor needs to be guessed in advance.

Assembling the left adjoint on arrows:

F (f)

is the unique map whose

G

-image completes the naturality square over

f

; functoriality follows by pasting squares and appealing to uniqueness.

In practice a left adjoint is found by solving the universal problem for each object separately; uniqueness then assembles the solutions into a functor. Barr & Wells run exactly this argument for free monoids — the universal property of $X^{*}$ alone forces $F$ to be a functor and $η$ to be natural⁹ — and state the general pointwise adjointness theorem: if for each $A$ there is an object $F (A)$ with $B (F (A), -) ≅ A (A, G (-))$ naturally, the assignment extends uniquely to a left adjoint.¹⁰ The adjoint functor theorem begins from this corollary: it gives conditions under which the required initial objects exist.

Everything dualizes. A right adjoint to $F : A \to B$ exists iff each comma category $(F \Rightarrow B)$ has a terminal object, a component $ε_{B} : F G (B) \to B$ through which every map $F (A) \to B$ factors uniquely.¹¹

Free and co-free solutions

Simmons frames the same idea as a universal solution to a problem; the vocabulary recurs throughout algebra.¹² Fix a functor $G : B \to A$ (often forgetful, but not necessarily). The free problem for an object $A \in A$ asks for an arrow $A \to G (S)$ comparing $A$ with a transported object; a $G$ -free solution is a uniform choice of

F (A) \in B, η_{A} : A \to (G \circ F) (A),

one per object $A$ , such that for each arrow $f : A \to G (S)$ there is a unique $f^{♯} : F (A) \to S$ with $G (f^{♯}) \circ η_{A} = f$ . This is word-for-word the initiality condition, stated without naming comma categories. Two theorems connect the two:

Every adjunction yields solutions. If $F ⊣ G$ with unit $η$ and counit $ε$ , then $(F, η, (\cdot)^{♯})$ is a $G$ -free solution and $(G, ε, (\cdot)^{♭})$ is an $F$ -co-free solution — the transposition formulas are the factorizations.¹³
Every solution yields an adjunction. Given a $G$ -free solution, the object assignment $F$ fills out to a functor left adjoint to $G$ , with $(\cdot)^{♯}$ as transposition and $η$ as unit. The proof is a chain of appeals to the uniqueness clause — Simmons counts some 8 or 9 — with no other ingredient.¹⁴

The dual co-free problem seeks $ε_{S} : (F \circ G) (S) \to S$ through which every $g : F (A) \to S$ factors as $g = ε_{S} \circ F (g^{♭})$ for a unique $g^{♭}$ ; co-free solutions correspond to right adjoints. Free monoids, groups, and vector spaces are the free constructions; indiscrete topologies and co-free algebras the co-free ones.

Vocabulary	Meaning	Equivalent to
$G$ -free solution at $A$	$η_{A} : A \to GF (A)$ , every $f$ factors uniquely	initial object of $(A \Rightarrow G)$
uniform $G$ -free solution	one such $η_{A}$ per object $A$	left adjoint $F ⊣ G$
$F$ -co-free solution at $S$	$ε_{S} : F G (S) \to S$ , every $g$ factors uniquely	terminal object of $(F \Rightarrow S)$
uniform $F$ -co-free solution	one such $ε_{S}$ per object $S$	right adjoint $F ⊣ G$

An adjunction can be recognized (hom-set bijection), manipulated (unit and counit), or built (universal arrows); the three descriptions suit the three tasks.

Leinster, Basic Category Theory, §2.3 opening — the everyday statement of the free vector space's universal property, diagram (2.7), and the claim that it is equivalent to $F ⊣ U$ with unit $η$ . ↩ ↩²
Leinster, §2.3, Definition 2.3.1 and Examples 2.3.3–2.3.4 — the comma category $(P \Rightarrow Q)$ , slice and coslice as special cases, the comma category $(A \Rightarrow G)$ , and the abuse of notation calling $f$ an object. ↩ ↩²
Leinster, §2.3, Lemma 2.3.5 — the unit map $η_{A}$ is an initial object of $(A \Rightarrow G)$ ; the proof identifies maps out of $(F (A), η_{A})$ with transposes. ↩
Leinster, Basic Category Theory, §2.2, Example 2.2.7 and the following remark — topological closure as left adjoint to the inclusion of closed subsets, with unit $A \subseteq Cl (A)$ initial among closed sets containing $A$ . ↩
Barr & Wells, Category Theory for Computing Science, §13.2, Definition 13.2.1 — adjointness defined by the universal mapping property of $η$ : each $f : A \to U B$ has a unique $g : F A \to B$ with $f = U g \circ η A$ . ↩
Barr & Wells, §13.2 after Definition 13.2.1, and §13.3, Proposition 13.3.6 — $η_{A}$ is a universal element of $Hom (A, U (-))$ ; $U$ has a left adjoint iff this functor has a universal element for every $A$ . ↩
Leinster, §2.3, Theorem 2.3.6 — one-to-one correspondence between adjunctions and natural transformations $η$ with every component initial; the uniqueness and existence of the counit both argued by initiality in $(G (B) \Rightarrow G)$ . ↩
Leinster, §2.3, Corollary 2.3.7 — $G$ has a left adjoint iff each $(A \Rightarrow G)$ has an initial object, with the objectwise construction of $F$ on arrows and its functoriality easily checked from uniqueness. ↩
Barr & Wells, §13.1, Propositions 13.1.2–13.1.4 — the universal mapping property of the free monoid, and the proof that the object assignment $X \mapsto F (X)$ becomes a functor with $η$ natural using only that property. ↩
Barr & Wells, §13.3, Theorem 13.3.5 (Pointwise Adjointness Theorem) — a natural equivalence $Hom (F A, -) ≅ Hom (A, U -)$ specified objectwise extends uniquely to a left adjoint functor. ↩
Leinster, §2.3, Exercise 2.3.9 — the dual of Corollary 2.3.7: right adjoints from terminal objects in the dual comma categories. ↩
Simmons, An Introduction to Category Theory, §5.5 and Definition 5.5.1 — free and co-free problems and solutions across a functor, defined in parallel columns; the mnemonic for each $f$ there is an $f^{♯}$ making the triangle commute. ↩
Simmons, §5.5, Theorem 5.5.2 — every adjunction provides a $G$ -free solution (via $F$ , $η$ , $(\cdot)^{♯}$ ) and an $F$ -co-free solution (via $G$ , $ε$ , $(\cdot)^{♭}$ ). ↩
Simmons, §5.5, Theorem 5.5.3 — a $G$ -free solution makes the object assignment $F$ a functor left adjoint to $G$ with $η$ as unit; the proof is a sequence of appeals to the uniqueness clause. ↩