A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably essentially injective if, for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the functor
\[ f^{*}\colon \mathsf{Hom}_{\mathcal{C}}(B,X)\to \mathsf{Hom}_{\mathcal{C}}(A,X) \]
given by precomposition by $f$ is essentially injective.
