I am trying to proof the existence of a right adjoint to the functor $F_*\colon \mathrm{Set}^D \to \mathrm{Set}^C$ given by precomposing with a functor $F\colon C \to D$, where $C$ and $D$ are small categories, using the dual version of the special adjoint functor theorem.
One of the ingredients for this is that $\mathrm{Set}^C$ is well-copowered (in the sense that $(\mathrm{Set}^C)^{\mathrm{op}}$ is well powered), which means--assuming I got my dualization right--that the collection of isomorphism classes of epimorphisms $F \to G$ in $(F \downarrow \mathrm{Set}^C)$ is (isomorphic to) a set.
This seems like a well known fact, but I couldn't find any proof of it and also failed to think of one by myself. Any hints or references to why this should be true?
If $\mathcal{S}$ is a category with cokernel pairs and $\mathcal{C}$ is a non-empty small category, then $\mathcal{S}$ is well-copowered if and only if $[\mathcal{C}, \mathcal{S}]$ is. The key observation is this: a morphism in $[\mathcal{C}, \mathcal{S}]$ is an epimorphism if and only if all its components are, because being an epimorphism is something that can be detected by cokernel pairs, and cokernel pairs in $[\mathcal{C}, \mathcal{S}]$ are computed componentwise if $\mathcal{S}$ has them.
The point is that the epimorphisms in $\mathbf{Set}$ are surjections, so there is an obvious cardinality bound on the quotients of any given set.