The Clowder Project

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably fully faithful on cores if it satisfies the conditions in Remark 9.8.4.1.2 and Remark 9.8.5.1.2, i.e.:

1. For all diagrams of the form
   

   with $\alpha $ and $\beta $ natural isomorphisms, if we have $\text{id}_{F}\mathbin {\star }\alpha =\text{id}_{F}\mathbin {\star }\beta $, then $\alpha =\beta $.

2. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism
   
   of $\mathcal{C}$, there exists a natural isomorphism
   
   of $\mathcal{C}$ such that we have an equality
   
   of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have
   \[ \beta =\text{id}_{F}\mathbin {\star }\alpha . \]