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Suppose we fix a universe $U$ and a U-small $U$-small category C. $C$. The regular Yoneda lemma gives us some locally small (not necessarily locally U-small?) functor category C'=[C^op,Sets] $C'=[C^{op},Sets]$ with a fully faithful embedding $C\rightarrow C'$ and the canonical bijection between Nat(F,Hom(-,x)) $Nat(F,Hom(-,x))$ and F(X). $F(X)$. Suppose we consider now, the U-small Yoneda lemma, that is, we look at [C^op,U-Sets]. $[C^{op},U-Sets]$. This is well-behaved since even though it is not U-small, Ob([C^op,U-Sets]) $Ob([C^{op},U-Sets])$ is still a set.
So the main question I have is: Are there any useful properties of the standard Yoneda lemma that we cannot reproduce with the U-small $U$-small Yoneda lemma for some U?$U$?