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