A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably fully faithful1 if the following equivalent conditions are satisfied:
- The $1$-morphism $f$ is corepresentably full (Definition 11.2.2.1.1) and corepresentably faithful (Definition 11.2.1.1.1).
-
For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor
\[ f^{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright ) \]
given by precomposition by $f$ is fully faithful.
1Further Terminology: Corepresentably fully faithful morphisms have also been called lax epimorphisms in the literature (e.g. in [Adámek–Bashir–Sobral–Velebil, On Functors Which Are Lax Epimorphisms]), though we will always use the name “corepresentably fully faithful morphism” instead in this work.
