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