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Stream: learning: questions

Topic: enrichment of functor categories

Jonas Frey (Jul 26 2026 at 00:20):

Given a small category $C$ and a locally small category $X$, is it true that $[C^{op},X]$ always has a canonical enrichment in $[C^{op},Set]$ given by $\underline{hom}(F,G)(c) = Nat(F/c,G/c)$?, where $F/c$ is the pullback of $F$ to $C/c$ ? If yes, what's a good reference for that?

I'm vaguely aware that under sufficient co/completeness assumptions on $X$ we get a situation with related enrichment, tensor, and cotensor structures, but I'm wondering about the enrichment alone.

Vincent Moreau (Jul 26 2026 at 19:50):

At least when $X = \operatorname{Set}$ this looks very much like the content of Exercise 8 of Chapter 1 in Sheaves in Geometry and Logic, about the internal hom of presheaf categories hence self-enrichment, see the screenshot attached:

It's quite interesting that it may be more general as you describe!