A simple question about enriched full and faithful functors

Question

Suppose that $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$ are $\mathcal{V}$-enriched categories and $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{E}$ are $\mathcal{V}$-functors.

Suppose further that both $G\circ F$ and $G$ are full and faithful in the enriched sense. Is it always the case that $F$ is also full and faithful?

It is true in the unenriched case, of course, and also in the examples I have in mind, but I am checking just in case something strange happens in general - which is quite often the case in the enriched world.

Answer 1

Yes. For every pair of objects in $C$, you have the diagram in $V$, $$C(X,Y)\to D(FX,FY)\to E(GFX,GFY)$$. As usual, if the composition and the second map are isomorphisms, then the first map is as well.