Category theory; homological algebra Adjunctions Theorem

Hom-set characterization of adjunctions

Builds on Adjoint functors Unit and counit Natural transformation

Statement

Let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ be functors.

Let $\varphi$ be an adjunction between $F$ and $G$ , with unit and counit $\eta$ and $\varepsilon$ . Then $\eta$ and $\varepsilon$ are natural transformations and satisfy $\varepsilon_{F c} \circ F(\eta_c) = \id_{F c}$ and $G(\varepsilon_d) \circ \eta_{G d} = \id_{G d}$ .
Conversely, let $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}_{\mathcal{D}}$ be natural transformations that satisfy these two triangle identities. Then $\varphi(g) = G(g) \circ \eta_c$ , for $g \colon F c \to d$ , is an adjunction. Its inverse is $f \mapsto \varepsilon_d \circ F(f)$ for $f \colon c \to G d$ .
These two constructions are inverse to each other.

Remarks

For part 1, naturality of $\varphi$ gives $\varphi(g) = G(g) \circ \eta_c$ and $\varphi^{-1}(f) = \varepsilon_d \circ F(f)$ . Applying these to identities gives the triangle identities. For part 2, the triangle identities are exactly what is needed for the two maps to be inverse to each other, and naturality is a direct check.

Example: for sets, $F = - \times Y$ and $G = (-)^Y$ , the unit sends $x \in X$ to the map $y \mapsto (x, y)$ . The counit is evaluation $Z^Y \times Y \to Z$ . The triangle identities then say that currying and evaluating undo each other.

Going deeper

For a student

The hom-set form is useful for recognizing adjunctions, since it is often a familiar bijection such as currying or the universal property of free objects. The unit and counit form is better for computation, because it is a finite set of equations between natural transformations. It also makes sense in any 2-category, so it is the form used to define adjunctions in general. A third equivalent form says that each $\eta_c$ is an initial object of the comma category $c \downarrow G$ .

Reading path

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Adjoint functors concept
Natural transformation concept
Unit and counit concept
Hom-set characterization of adjunctions statement

Tag	`4KCECXYW81`
URI	`m2://4KCECXYW81`
Kind	statement, theorem
Writing	W0 stub
MSC	18A40
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About	Adjoint functors, Unit and counit
Uses	Adjoint functors, Unit and counit, Natural transformation
Changed	2 November 2026