Reading here, https://en.wikipedia.org/wiki/Full_and_faithful_functors, I found that a functor $$F:{\mathcal C}\rightarrow {\mathcal D}$$ is faithful if the function
$$F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))$$
is injective for every $X$ and $Y$.
What is this function?
I suppose that it is
$$F_{X,Y}(f)(x) = Ff(x)\ \ \ \ \ \ \forall\ x\in F(X)$$
Yes. Note however that you need to assume that your categories are ``locally small'', so that indeed $\hom_{\mathcal{C}}(X,Y)$ and $\hom_{\mathcal{D}}(FX,FY)$ form actual sets.