Suppose we fix a universe $U$ and a $U$-small category $C$. The regular Yoneda lemma gives us some locally small (not necessarily locally U-small?) functor category $C'=[C^{op},Sets]$ with a fully faithful embedding $C\rightarrow C'$ and the canonical bijection between $Nat(F,Hom(-,x))$ and $F(X)$. Suppose we consider now, the U-small Yoneda lemma, that is, we look at $[C^{op},U-Sets]$. This is well-behaved since even though it is not U-small, $Ob([C^{op},U-Sets])$ 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 Yoneda lemma for some $U$?