By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I am interested in knowing to what extent this is true for enriched categories.
Question. Is there an example of a non-thin symmetric closed monoidal category $\mathscr V$ and a $\mathscr V$-category $C$ satisfying the following two conditions?
- The $\mathscr V$-enriched presheaf category $[C^{\text{op}}, \mathscr V]$ exists.
- $C$ is not Morita-equivalent to a small $\mathscr V$-category (i.e. there does not exist a small $\mathscr V$-category $C'$ such that $[C^{\text{op}}, \mathscr V] \simeq [(C')^{\text{op}}, \mathscr V])$.
Background. Conceptually, the obstruction to such examples is the absence of arbitrary large limits in $\mathscr V$ (needed to form $\mathscr V$-enriched functor categories). Indeed, if $[C^{\text{op}}, \mathscr V]$ exists, then for every pair of $\mathscr V$-functors $F, G : C^{\text{op}} \to \mathscr V$, we have an end $$\int_{c \in C^{\text{op}}} \mathscr V(Fc, Gc)$$ in $\mathscr V$, which in general is a large limit. Since $\mathscr V$ is itself a large category, if it admits arbitrary large limits (in fact, large powers suffice), it must be thin (which is why this case is excluded). Therefore, if it is possible to construct large powers from such ends (assuming $C$ is not "Morita-small"), the question above would be answered in the negative. However, it is not clear to me whether this is possible in general, leaving open the possibility of a (nontrivially) large $\mathscr V$-category admitting a $\mathscr V$-enriched presheaf category.
(It does seem to be true that we can construct large powers of arbitrary objects in $\mathscr V$ from such ends in special cases, e.g. when $C$ is discrete, by taking $F$ to be constant on the unit of $\mathscr V$, and $G$ to be constant on a fixed object of $\mathscr V$.)
