CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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If A is small, then [A,C] is complete or cocomplete whenever C is respectively complete or cocomplete

Statement

Assume Choice. If $A$ is small and $C$ is complete, then $[A, C]$ is complete. If $C$ is cocomplete, then $[A, C]$ is cocomplete.

Facts & Assumptions

Given: A small $A$ and the indicated completeness or cocompleteness of $C$ .

[L1]

Functor-category limits and colimits are computed pointwise when the pointwise choices exist (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise).

[F1]

Completeness and cocompleteness mean existence of every small limit and colimit (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

Choice permits simultaneous selections from a set-indexed family of nonempty sets (The Axiom of Choice).

Proof

technique · direct corollary

1.1

Let $D : J \to [A, C]$ be small. If $C$ is complete, [F1] supplies a limit of $j \mapsto D (j) (a)$ for every $a$ . Since $A$ is small, [F2] selects these pointwise limits simultaneously.

F1F2

2.1

By [L1], the selected objects form a limit of $D$ . Since $D$ was arbitrary, [F1] says the functor category is complete.

L1F1step 1.1

3.1

If $C$ is cocomplete, make the same set-indexed choices of pointwise colimits and apply the colimit clause of [L1]. This proves cocompleteness, with no assertion for a large source $A$ .

L1F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

E. Riehl, Category Theory in Context, Corollary 3.3.3 (standard reference, not scraped)