A functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably full on cores if, for each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition by $F$ functor
\[ F_{*}\colon \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright )\webright ) \]
is full.
