I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can probably assume that $\mathcal C$ and $\mathcal D$ are as nice as you want but I think it suffices for them to be locally small well-powered and co-well-powered (so that the subobjects or quotient objects of a given object form a set, thank you, Zhen Lin). I believe that for any object $X$ in $\mathcal C$ such an $\mathcal F$ will induce a bijection between the subobjects of $X$ and the quotient objects of $\mathcal F(X)$ by simply sending the class of a monomorphism $\alpha$ to the class of the epimorphism $\mathcal F(\alpha)$. This appears semi-obvious and a detailed proof is easy to give but it is a bit longer than I'd like to put in my paper, since the subject is something else completely. So my question is: what would be a reference for a statement of this form, preferably in a textbook on category theory.
For context, I'd like to apply this when $\mathcal C$ is the category of finite posets, $\mathcal D$ is the category of finite distributive lattices and $\mathcal F$ is the functor sending a poset to the the lattice of its order ideals. Also, the dual statement is, of course, the even more intuitive claim that a category equivalence induces bijections between sets of subobjects. A reference for that would also be nice but I'm hoping for the above precise form.
