A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably fully faithful[1] if the following equivalent conditions are satisfied:
- The $1$-morphism $f$ is representably faithful (Definition 9.1.1.1.1) and representably full (Definition 9.1.2.1.1).
-
For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor
\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright ) \]
given by postcomposition by $f$ is fully faithful.
