A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably essentially injective if, for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, 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 essentially injective.
