Units, Counits, and the Triangle Identities

The hom-set definition of an adjunction carries one bijection for every pair of objects, plus naturality in both variables. Transposing identity maps produces two natural transformations, the unit and the counit; they satisfy two equations, the triangle identities. The main theorem is that any pair of natural transformations satisfying those equations arises from exactly one adjunction, so the bijection can be discarded and rebuilt at will.

The unit and the counit

Fix an adjunction $F ⊣ G$ with $F : A \to B$ and $G : B \to A$ . The transpose bijection

B (F (A), B) ≅ A (A, G (B))

accepts any map on either side. Feed it the maps that exist canonically: identities. Taking $B = F (A)$ , the identity $1_{F (A)}$ lives in the left-hand set, and its transpose lives in the right-hand one; dually with $A = G (B)$ .

The unit component

u_{A} = η_{A}

is the transpose of the identity on

F (A)

; the counit component

e_{B} = ε_{B}

is the transpose of the identity on

G (B)

. Each lives one level down or up from an identity map.

Naturality is not an extra hypothesis; it follows from the naturality of the adjunction itself. For any $k : A^{'} \to A$ in $A$ , applying the transpose axioms to $1_{F (A)}$ and $F (k)$ gives $GF (k) \circ η_{A^{'}} = η_{A} \circ k$ , the naturality square for $η$ ; the counit is dual.¹

The free vector space

Take the free–forgetful adjunction $F ⊣ U$ between $Set$ and $Vect_{k}$ , where $F (S)$ is the vector space of formal $k$ -linear combinations of elements of $S$ .²

Unit $η_{S} : S \to U F (S)$ sends an element $s$ to itself, viewed as a basis vector — the insertion of generators into the free object.
Counit $ε_{V} : F U (V) \to V$ sends a formal linear combination $\sum_{v} λ_{v} v$ of vectors of $V$ to its actual value in $V$ — formal evaluation.

The two components are very different in size. The unit is a modest inclusion. The counit's domain $F U (V)$ has one basis vector for every element of $V$ : for $V = R^{2}$ it is an uncountably infinite-dimensional space, collapsed by $ε_{V}$ onto the plane.² Units are typically embeddings of a thing into its freely generated hull; counits typically evaluate or project.

Recovering the bijection

The unit and counit are only the transposes of identities, yet they generate the entire adjunction.

By naturality, $\overset{g}{ˉ} = \overline{g \circ 1_{F (A)}} = G (g) \circ \overline{1_{F (A)}} = G (g) \circ η_{A}$ , and dually.³ So the entire bijection is: apply the functor, then compose with the unit (or counit).

The triangle identities

Since $η$ and $ε$ determine the transpositions, the equations $\overset{ˉ}{\overset{ˉ}{f}} = f$ and $\overset{ˉ}{\overset{g}{ˉ}} = g$ must translate into equations on $η$ and $ε$ alone. Specializing the lemma to $f = η_{A}$ and $g = ε_{B}$ gives the translation.

The two triangle identities: going up with

F (u_{A})

then down with

e_{F (A)}

is the identity on

F (A)

, and symmetrically for

G

. Here

u

is the unit

η

and

e

the counit

ε

For the proof, apply the recovery lemma to the definition of the unit: $η_{A} = \overline{1_{F (A)}}$ transposes back to $1_{F (A)} = \overset{η}{ˉ}_{A} = ε_{F (A)} \circ F (η_{A})$ , the first triangle; the second is dual.⁴ Simmons derives the same pair as a corollary of the transposition formulas and calls them the identities that, in appropriate circumstances, determine the adjunction.⁵

Worked example (Triangle identities for the free monoid). Take the free–forgetful adjunction $F ⊣ U$ between $Set$ and $Mon$ , where $F (X) = X^{*}$ is the monoid of words over $X$ under concatenation. Fix $X = {a, b}$ .

Unit $η_{X} : X \to U F (X)$ sends $a \mapsto ⟨ a ⟩$ and $b \mapsto ⟨ b ⟩$ . Transposing a function $u : X \to U (M)$ evaluates a word: for $M = (N, +, 0)$ with $u (a) = 1$ , $u (b) = 2$ , the transpose gives $\overset{u}{ˉ} ⟨ abab ⟩ = 1 + 2 + 1 + 2 = 6$ .
Counit $ε_{M} : F U (M) \to M$ multiplies a word of elements out: $ε_{M} ⟨ 35 ⟩ = 3 + 5 = 8$ .

First triangle, $ε_{F (X)} \circ F (η_{X}) = 1_{F (X)}$ , on the word $⟨ ab ⟩$ : $F (η_{X})$ replaces each letter by its one-letter word, producing the length-two word $⟨ ⟨ a ⟩ ⟨ b ⟩ ⟩$ in $F U F (X)$ , and $ε_{F (X)}$ concatenates its entries back to $⟨ a ⟩ ⟨ b ⟩ = ⟨ ab ⟩$ . Second triangle, $U (ε_{M}) \circ η_{U (M)} = 1_{U (M)}$ , on $m \in N$ : $η$ forms the one-letter word $⟨ m ⟩$ and $ε_{M}$ evaluates it to $m$ . Both composites are the identity.

The identities are sometimes called the zig-zag equations, after their string-diagram form. Drawing a functor as a vertical string, the unit as a cap where an $F$ -string and a $G$ -string are born, and the counit as a cup where they annihilate, each identity says a zig-zag in a string can be pulled straight.⁶

The zig-zag form of the first triangle identity: a bent

F

-string (born at a unit cap, absorbed at a counit cup) equals the straight string. Caps are units, cups are counits.

Adjunctions from a unit–counit pair

A unit–counit pair, subject only to the triangle identities, is a complete presentation of the adjunction.

The inverse-pair check in the unit-counit theorem: naturality of the counit

e

makes the square commute, and the first triangle identity collapses the left edge, giving the outer composite

= g

The square commutes by naturality of $ε$ applied to $g$ , and the left triangle is the first triangle identity. Reading the outer boundary, $ε_{B} \circ F G (g) \circ F (η_{A}) = g \circ ε_{F (A)} \circ F (η_{A}) = g \circ 1 = g$ . Dually $\overset{ˉ}{\overset{ˉ}{f}} = f$ . Naturality of the resulting bijection in $A$ and $B$ follows from functoriality of $F$ and $G$ together with naturality of $η$ and $ε$ ; and the unit of the resulting adjunction is $\overline{1_{F (A)}} = G (1_{F (A)}) \circ η_{A} = η_{A}$ , as required.⁷⁸ $■$ Simmons proves the same theorem by splitting the naturality axiom into four one-variable fragments and showing various subsets suffice; the useful residue of that analysis is that in practice one rarely verifies full two-variable naturality — the unit–counit route replaces it with two equations between composites of natural transformations.⁹

The data dictionary

Three packages of data now describe the same thing.

Package	Data	Conditions	Best for
hom-set bijection	$B (F A, B) ≅ A (A, GB)$ for all $A, B$	naturality in $A$ and $B$	recognizing adjunctions in examples
unit and counit	$η : 1 \Rightarrow GF$ , $ε : F G \Rightarrow 1$	two triangle identities	proving theorems, composing adjunctions
universal arrows	$η_{A} : A \to GF (A)$ initial for each $A$	initiality in $(A \Rightarrow G)$	building an adjoint objectwise

The third package, universal arrows, builds the adjoint one object at a time. The middle package has a feature the others lack: it never mentions elements of hom-sets, only functors and natural transformations, so it ports to any 2-categorical setting where bijection of hom-sets makes no sense.

Order-theoretic adjunctions

Between posets, adjunctions collapse to something checkable at a glance. Let $f : A \to B$ and $g : B \to A$ be order-preserving maps between ordered sets, viewed as functors between thin categories. Each hom-set has at most one element, so the bijection condition is a biconditional and naturality is automatic:

f ⊣ g \leftrightarrow (f (a) \leq b \leftrightarrow a \leq g (b) for all a, b) .

The unit and counit become inequalities, and the triangle identities become vacuous (any two parallel maps in a poset are equal):¹⁰

Unit. $a \leq g f (a)$ for all $a \in A$ — applying $f$ then $g$ can only move up.
Counit. $f g (b) \leq b$ for all $b \in B$ — applying $g$ then $f$ can only move down.

By the unit–counit theorem, the biconditional holds iff both inequalities hold, a fact provable directly in three lines but now an instance of a theorem about adjunctions in general.

For example, let $X$ be a topological space, $P (X)$ its poset of subsets and $C (X)$ its poset of closed subsets, both ordered by inclusion. Closure and inclusion form an adjunction $Cl ⊣ i$ :

Cl (A) \subseteq B \leftrightarrow A \subseteq B (A \subseteq X, B closed) .

The unit is $A \subseteq Cl (A)$ (a set sits inside its closure) and the counit is $Cl (B) \subseteq B$ (a closed set already contains its closure — here equality).¹⁰ The closure operation, usually presented by axioms, is an adjoint.

Two cautions

An equivalence's isomorphisms need not be its unit and counit

An equivalence of categories $(F, G, η, ε)$ consists of natural isomorphisms $η : 1 \Rightarrow GF$ and $ε : F G \Rightarrow 1$ , with no triangle condition. In any equivalence $F ⊣ G$ ; but the given $η$ and $ε$ need not be the unit and counit of that adjunction, precisely because they may fail the triangles. They can always be adjusted to a pair that satisfies them.¹¹

The unit and counit need not be isomorphisms

How far they fail measures how far the adjunction is from an equivalence. The failure is itself structured:

The right adjoint $G$ is full and faithful iff the counit $ε$ is a natural isomorphism. Such an adjunction is called a reflection; the abelianization adjunction $Ab ⇄ Grp$ is one, with $Ab$ a reflective subcategory of $Grp$ .¹²
On the full subcategories where the unit (respectively counit) is an isomorphism, every adjunction restricts to an equivalence — the fixed points of the adjunction.¹³

For the free vector space adjunction, neither transformation is invertible: the unit $S \to U F (S)$ is far from surjective ( $U F (S)$ contains all formal sums), and the counit collapses an enormous space. The adjunction relates $Set$ and $Vect_{k}$ without identifying them.

The unit component $η_{A} : A \to GF (A)$ has a universal property on its own, and a family of such universal arrows rebuilds the left adjoint.

Simmons, An Introduction to Category Theory, §5.4, Lemma 5.4.2 — naturality of the unit and counit, proved from the naturality fragment $([)$ of the adjunction axioms instantiated twice. ↩
Leinster, Basic Category Theory, §2.2, Example 2.2.1 — unit and counit of the free–forgetful adjunction for vector spaces; the counit evaluates formal sums, and $F U (V)$ is uncountably infinite-dimensional even for $V = R^{2}$ . ↩ ↩²
Leinster, §2.2, Lemma 2.2.4 — $\overset{g}{ˉ} = G (g) \circ η_{A}$ and $\overset{ˉ}{f} = ε_{B} \circ F (f)$ ; the unit and counit appear to know only the transposes of identities yet determine the whole adjunction. Also Simmons §5.4, Lemma 5.4.3. ↩
Leinster, §2.2, Lemma 2.2.2 and Remark 2.2.3 — the triangle identities as commuting diagrams in the functor categories $[A, B]$ and $[B, A]$ , with the componentwise form (2.4). ↩
Simmons, §5.4, Corollary 5.4.4 — the identities $ε_{F A} \circ F (η_{A}) = 1_{F A}$ and $G (ε_{S}) \circ η_{GS} = 1_{GS}$ obtained by specializing the transposition formulas to the unit and counit. ↩
Leinster, §2.2, Remark 2.2.9 — string diagrams for natural transformations; the unit and counit as cap and cup, and the triangle identities as pulling the string straight. ↩
Leinster, §2.2, Theorem 2.2.5 and Corollary 2.2.6 — one-to-one correspondence between adjunctions and unit–counit pairs satisfying the triangle identities; the commuting-square argument for $\overset{ˉ}{\overset{g}{ˉ}} = g$ . ↩
Simmons, §5.4, Theorem 5.4.5 — a pair of natural transformations satisfying the Corollary 5.4.4 identities is the unit and counit of a unique adjunction, with the transpositions rebuilt by the recovery formulas. ↩
Simmons, §5.3 — the naturality requirement decomposed into the four one-variable fragments $(♯ ↑), (♯ ↓), (♭ ↑), (♭ ↓)$ , and Lemma 5.3.2 listing which pairs of fragments suffice for an adjunction. ↩
Leinster, §2.2, Example 2.2.7 — adjunctions between ordered sets: the biconditional, the unit and counit as inequalities, the vacuous triangle identities, and topological closure as a left adjoint to inclusion of closed sets. ↩ ↩²
Leinster, §2.2, Remark 2.2.8 with Exercise 2.3.10 — in any equivalence $(F, G, η, ε)$ the functor $F$ is left adjoint to $G$ , but the given $η, ε$ need not satisfy the triangle identities, so they need not be the unit and counit. ↩
Leinster, §2.2, Exercise 2.2.12 — the right adjoint is full and faithful iff the counit is an isomorphism; such adjunctions are reflections, e.g. $Ab$ reflective in $Grp$ (Example 2.1.3(c)–(d)). ↩
Leinster, §2.2, Exercise 2.2.11 — every adjunction restricts canonically to an equivalence between the full subcategories on which the unit (resp. counit) is an isomorphism. ↩

The unit and the counit

The free vector space

Recovering the bijection

The triangle identities

Adjunctions from a unit–counit pair

The data dictionary

Order-theoretic adjunctions

Two cautions

An equivalence's isomorphisms need not be its unit and counit

The unit and counit need not be isomorphisms

Footnotes