The Clowder Project

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably fully faithful¹ if the following equivalent conditions are satisfied:

1. The $1$-morphism $f$ is representably faithful (Definition 11.1.1.1.1) and representably full (Definition 11.1.2.1.1).
2. 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.