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