Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors

Let $F:\mathcal C\to\mathcal D$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor). For each $A,B$, it induces

$F_{A,B}:\mathcal C(A,B)\to\mathcal D(FA,FB),\qquad f\mapsto Ff.$

The functor is faithful when every $F_{A,B}$ is injective, full when every $F_{A,B}$ is surjective, and fully faithful when every $F_{A,B}$ is bijective (Injection, surjection, bijection).

It is essentially surjective when every object $D$ of $\mathcal D$ is isomorphic to some $FC$ (Isomorphism, groupoid, and connected category). It is split essentially surjective when the data include, for every $D$, a specified object $C_D$ and specified isomorphism $\varepsilon_D:FC_D\to D$. The word split records these witnesses and is strictly stronger as data than their mere existence.