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 11.8.7.1.2 and Remark 11.8.8.1.2, i.e.:

1. 1.

   For all diagrams of the form

   
   if $\alpha $ and $\beta $ are natural isomorphisms and we have
   \[ \alpha \mathbin {\star }\operatorname {\mathrm{id}}_{F}=\beta \mathbin {\star }\operatorname {\mathrm{id}}_{F}, \]

   then $\alpha =\beta $.

2. 2.

   For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$ 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
   \[ \beta =\alpha \mathbin {\star }\operatorname {\mathrm{id}}_{F}. \]