Suppose we are in presence of a strong enough axiom of choice (e.g., choice for conglomerates). I know that any category has a skeleton, but I would like to know if I can choose a skeleton which contains some distinguished objects.
Let $C$ be a category and suppose that I have a distinguished class $D$ of pairwise non-isomorphic objects of $C$. Can I "complete" $D$ to a skeleton of $C$?
If the above question has a positive answer, suppose I have another class $D'$ of pairwise non-isomorphic objects in $C$ (which has trivial intersection with $D$) such that any object of $D'$ is isomorphic exactly to one object of $D$ and this exhausts all of $D$. Also, for any object $X$ in $D'$ we have a distinguished isomorphism $\phi_X$ between $X$ to the unique object in $D$ isomorphic to $X$.
Can I find a skeleton $S$ of $C$ and an equivalence of categories $F:C\to S$ such that $F(\phi_X)$ is an identity for all $X\in D'$?