it is faithful; that is, for any pair of objects$x,y\in C$ the component function$F \colon C(x,y) \to D(F x,F y)$ between hom-sets is injective, and

it is full on isomorphisms, meaning that for any pair of objects$x,y\in C$ the function$F \colon Iso_C(x,y) \to Iso_D(F x, F y)$ to the set of isomorphisms between them is surjective (hence bijective), where $Iso_C(x,y)$.

More generally, a 1-morphism$f \colon C\to D$ in any 2-category$K$ is called a pseudomonic morphism if the square analogous to that below is a 2-pullback, or equivalently if $K(X,C)\to K(X,D)$ is a pseudomonic functor for any $X$.

A functor $F \colon C \to D$ is pseudomonic if and only if the commuting square

$\array{
C &\overset{Id}{\longrightarrow}& C
\\
\Big\downarrow\mathrlap{^{Id}}
&&
\Big\downarrow\mathrlap{{}^F}
\\
C &\underset{F}{\longrightarrow}& D
}$

An interesting example of the notion appears in the context of Joyal’s combinatorial species of structures.

A combinatorial species is a functor from the category$Bij$ of finite sets and bijections between them to Set, and the functors that are obtained by taking left Kan extensions of species along the subcategory-inclusion $I \colon Bij \to Set$ are called analytic functors. Now taking left Kan extensions along $I$ is pseudomonic, and this implies that the coefficients of an analytic functor are unique up to isomorphism.

Arguably, pseudomonic functors are precisely the functors for which it makes sense to say that $A$ is uniquely determined by $F A$ up to unique isomorphism. However, we do not really need faithfulness for this; bijectivity on isos suffices.