Yoneda lemma for monoidal categories 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, 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 again assumes "smallness". In these answers, is smallness a necessary assumption?
 A: The Yoneda lemma is a purely formal result that does not require any size assumptions.  For any closed symmetric monoidal category $\mathbf{V}$, any $\mathbf{V}$-category $C$, any object $A\in C$, and any functor $F:C\to \mathbf{V}$, there is an isomorphism
$$ [C,\mathbf{V}](よ^A,F)  \cong F(A). $$
Here $よ^A$ denotes the hom-functor $C(A,-)$ and $[C,\mathbf{V}]$ denotes the $\mathbf{V}$-enriched hom-category.  It is true that one needs $\mathbf{V}$ to have limits of the size of $C$ in order for $[C,\mathbf{V}]$ to exist as a $\mathbf{V}$-category, but even if this fails, the statement is true and provable in the following sense: if we write down the diagram whose limit would, if it existed, be the LHS, then the RHS is a limit of that diagram.
It is even possible to formulate and prove versions of the Yoneda lemma that do not require $\mathbf{V}$ to be closed or symmetric, and even that allow it to be a multicategory rather than a monoidal category.  See, for instance, Lemma 5.29 of my paper enriched indexed categories, or proposition 8.2 of my paper with Richard Garner, enriched categories as a free cocompletion for the bicategorical case.
