PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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An adjunction induces postcomposition and precomposition adjunctions on legitimate functor categories

Statement

Let $F ⊣ G : D \to C$ . Whenever the indicated functor categories are legitimate:

postcomposition gives an adjunction $F_{*} : [J, C] ⇄ [J, D] : G_{*}$ ;
precomposition gives an adjunction $G^{*} : [C, E] ⇄ [D, E] : F^{*}$ .

If $J$ is small and $C, D$ are locally small, the functor categories in clause 1 are locally small.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ , and categories for which the displayed functor categories are formed.

[F1]

For a small source category, functors and natural transformations form the functor category (Functor category $[C, D]$ ).

[F2]

If the source is small and the target is locally small, the resulting functor category is locally small (If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L1]

The triangle identities are $(ε F) (F η) = 1_{F}$ and $(G ε) (η G) = 1_{G}$ (Adjunction by unit, counit, and the triangle identities).

Proof

technique · direct

1.1F1L1

Postcomposition sends $H : J \to C$ to $F H$ and $K : J \to D$ to $G K$ . Whiskering $η$ and $ε$ gives unit components $η H : H \Rightarrow G F H$ and counit components $ε K : F G K \Rightarrow K$ .

1.2F1L1

Precomposition sends $H : C \to E$ to $H G$ and $K : D \to E$ to $K F$ . Whiskering now gives the unit $H η : H \Rightarrow H G F$ and counit $K ε : K F G \Rightarrow K$ , so $G^{*} ⊣ F^{*}$ .

2.1step 1.1L1

The two triangle identities hold at every object of $J$ by [L1], hence hold as equalities of natural transformations. Therefore $F_{*} ⊣ G_{*}$ .

3.1step 1.2F2L1∎

Again the triangle identities are the images under $H$ and $K$ of those in [L1]. The size assertion follows from [F2]; the componentwise unit-counit construction itself uses only legitimate functors and natural transformations.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Proposition 4.3.6 (standard reference, not scraped)
Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter IV (standard reference, not scraped)