Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably full on cores if, for each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the postcomposition by $F$ functor
is full.
In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably full on cores if, for each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and each natural isomorphism
Is there a characterisation of functors representably full on cores?
This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
