How sensitive is the Yoneda lemma to set-theoretic subtleties?
Solution 1:
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 $\hom(\hom(X,-),F) \cong F(X)$.