Is there a name for functors $F : \mathcal{C} \to \mathcal{D}$ with the property that for all $A,B \in \mathrm{Ob}(\mathcal{C})$ the map $F : \mathrm{Isom}(A,B) \to \mathrm{Isom}(F(A),F(B))$ is an isomorphism? In other words, for every isomorphism $g : F(A) \to F(B)$ there is a unique isomorphism $f : A \to B$ with $g=F(f)$.
For example, the free functor $\mathsf{Set} \to \mathsf{Set}_*$, $X \mapsto (X+\{\star\},\star)$ has this property. From this example one can also see that $F$ does not have to be fully faithful.
Also note that $F$ does not have to be conservative, faithful, or full in general.
Such functors are precisely the pseudomonomorphisms in the 2-category of categories.
(A pseudomonomorphism is a morphism $f : X \to Y$ in a bicategory such that $$\require{AMScd} \begin{CD} X @= X \\ @| @VV{f}V \\ X @>>{f}> Y \end{CD}$$ is a bicategorical pullback square.)