CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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A right adjoint preserves ends and a left adjoint preserves coends

Statement

Let $T : C^{o p} \times C \to D$ be a functor and let $F ⊣ G$ be an adjunction with $F : D^{'} \to D$ and $G : D \to D^{'}$ (Adjunction by unit, counit, and the triangle identities).

If $T$ has an end (The end and the coend of a functor $C^{o p} \times C \to D$ ), then $G$ carries it to an end of $G T$ . If $T$ has a coend and $H ⊣ K$ is an adjunction with $H : D \to E$ , then $H$ carries that coend to a coend of $H T$ .

No smallness hypothesis on $C$ is imposed, because the published preservation theorem imposes none: it applies at every indexing category for which the diagram and cone categories are legitimate, and $Tw (C)$ is one such whenever $T π$ is a diagram at all.

Facts & Assumptions

Given: A functor $T$ on $C^{o p} \times C$ and an adjunction whose right or left half is applied to it.

[F1]

An adjunction $F ⊣ G$ consists of functors $F, G$ with unit and counit satisfying the triangle identities $(ε F) \circ (F η) = 1_{F}, (G ε) \circ (η G) = 1_{G} .$ The direction $F ⊣ G$ means that $F$ is left adjoint to $G$ and $G$ is right adjoint to $F$ (Adjunction by unit, counit, and the triangle identities).

[L2]

If a diagram $D : J \to D$ has a limit $(L, λ)$ and $F ⊣ G$ , then $(G L, G λ)$ is a limit of $G D$ . Thus $G$ preserves every limit that exists, for arbitrary indexing categories for which the displayed diagram and cone categories are legitimate. (Right adjoints preserve every limit that exists).

[L3]

If $F$ is a left adjoint and a diagram has a colimit, then applying $F$ to a colimiting cocone produces a colimit of the composite. Thus left adjoints preserve every colimit that exists. (Left adjoints preserve every colimit that exists).

[L1]

If $F$ preserves $Tw (C)$ -limits and $T$ has an end, then $F$ of that end is an end of $F T$ , so a functor preserving twisted-arrow limits preserves ends; dually a functor preserving $Tw (C)^{o p}$ -colimits carries a coend to a coend (A functor preserving twisted-arrow limits preserves ends, and dually for coends).

[F2]

An end of $T$ is a terminal object of the category of wedges over $T$ and a coend an initial object of the category of cowedges under $T$ ; in short, an end is a terminal wedge and a coend an initial cowedge (The end and the coend of a functor $C^{o p} \times C \to D$ ).

Proof

technique · direct

1.1

By [F1] the functor $G$ of the adjunction $F ⊣ G$ is a right adjoint, and by [L2] a right adjoint preserves every limit that exists, at arbitrary legitimate indexing categories. In particular it preserves $Tw (C)$ -limits, and no size hypothesis on $C$ is used to say so.

F1L2

2.1

So $G$ satisfies the hypothesis of [L1] at the indexing category $Tw (C)$ , and therefore carries an end of $T$ to an end of $G T$ .

F2L1L2step 1.1

3.1

Dually, by [L3] a left adjoint preserves every colimit that exists, hence preserves $Tw (C)^{o p}$ -colimits, and by the coend clause of [L1] it carries a coend of $T$ to a coend of $H T$ .

F2L1L3step 1.1∎

Remarks

The corollary is stated for a right adjoint and a left adjoint separately because the two halves of an adjunction do different things here: the right adjoint is the one that preserves the limit computing an end, and the left adjoint the one that preserves the colimit computing a coend. Applying the wrong half of an adjunction to the wrong universal object gives no information at all.

The hom-functor case is the one used most often below and is recorded separately as The hom-functor turns a coend into an end and carries an end to an end, because the covariant hom-functor turns a coend into an end rather than preserving a coend, and that change of shape is not an instance of the present corollary.

Depends on

Used by

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Dependency tree · two levels

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Sources

F. Loregian, (Co)end Calculus (arXiv:1501.02503v7), Theorem 1.2.7 (standard reference, not scraped)