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In it's usual form, the Yoneda lemma cannot apply to categories that are not locally small, because, for the functor of points to have codomain $\mathsf{Set}$, the hom-sets must actually be sets.

However, it seems as though one could use the same definition to create a functor from a locally-large category to some category of proper classes, and obtain much the same result.

Are there any issues that could arise in such an approach? Most sources on category theory gloss over set-theoretic subtleties, suggesting that they are ultimately not important, but my intuition is not very good on issues like this.

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Set-theoretic foundations really don't matter as for the Yoneda Lemma, since its proof is entirely formal. If $U$ is any Grothendick universe, then the Yoneda Lemma holds for $U$-categories: If $F : C \to U\mathsf{Set}$ is an $U$-functor and $X \in C$, then $\hom(\hom(X,-),F) \cong F(X)$.

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Martin: could you perhaps indicate what you mean by "U-category"? I am familiar with "U-small/moderate category". –  magma Jan 10 at 8:30
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The classical reference is SGA 4, Exp. 1. A $U$-category is a category whose hom-sets are $U$-small, i.e. are isomorphic to elements of $U$. –  Martin Brandenburg Jan 10 at 9:27

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