I am looking at the Yoneda lemma trying to see where the assumption of "locally small" really comes in. Obviously in order to define a functor to the category sets using $Hom$-spaces we need our $Hom$-spaces to be sets. However if we consider a enriched-category, enriched over some non-locally small monoidal category **M**, then for any element of the category, our $Hom$-sets give us a functor $Hom(A,-)$ to **M**. 

In particlar, in the statement
$$
Hom(Hom(-,A),F) \simeq F(A),
$$
where $F$ is a set-valued functor, where does the assumption of "smallness" play a role.  

In the answer to this [question][1], it is stated that the category of sets can be replaced by any Grothendieck universe $U$. However, the definition of a Grothendieck universe assumes that $U$ is a set. Moreover, the [enriched Yoneda lemma][2] again assumes "smallness". In these answers, is smallness a necessary assumption?


  [1]: https://math.stackexchange.com/questions/633386/how-sensitive-is-the-yoneda-lemma-to-set-theoretic-subtleties
  [2]: https://ncatlab.org/nlab/show/enriched+Yoneda+lemma