Concrete functor between concrete categories

I'm almost certain the answer is "Nothing interesting", but I'm asking the question anyway. Let $(A,U), (B,V)$ be two concrete categories over some category $X$.

Assume we have a concrete faithful functor $F: A\to B$ of which we know that it isn't full, nor essentially surjective (aka "isomorphism dense"). We also know that $X,A,B$ have all finite products and coproducts and that all of $F,U,V$ commute with these. What can we say about $F$, or $Im(F)$ ?

If I make things a bit more precise and add that $X= \bf{Set}^f$ (category of finite sets), is there anything interesting to say ?

I'm sorry if this is too vague and too imprecise what I'm asking for, but I simply want to know if in this situation there are general interesting results that may tell me more about the situation.

• When you say "commute with finite (co)products", do you mean "preserve finite (co)products"? – Arnaud D. Jun 9 '17 at 15:49
• Yes sorry, what I meant was : for all objetcs $A,B$ in the appropriate category, $G(A\times B) = G(A)\times G(B)$ (or more generally, replace $=$ by $\simeq$, where the isomorphism is natural in both $A$ and $B$) – Max Jun 9 '17 at 16:05