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](https://math.stackexchange.com/questions/633386/how-sensitive-is-the-yoneda-lemma-to-set-theoretic-subtleties), 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](https://ncatlab.org/nlab/show/enriched+Yoneda+lemma) again assumes "smallness". In these answers, is smallness a necessary assumption?



Yoneda lemma for monoidal categories
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](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html), or proposition 8.2 of my paper with Richard Garner, [enriched categories as a free cocompletion](https://arxiv.org/abs/1301.3191) for the bicategorical case.


