The Clowder Project

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably fully faithful on cores if it satisfies the conditions in Remark 9.8.7.1.2 and Remark 9.8.8.1.2, i.e.:

1. For all diagrams of the form
   

   if $\alpha $ and $\beta $ are natural isomorphisms and we have

   \[ \alpha \mathbin {\star }\text{id}_{F}=\beta \mathbin {\star }\text{id}_{F}, \]

   then $\alpha =\beta $.

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