The Clowder Project

In detail, $f$ is representably fully faithful if the conditions in Remark 11.1.1.1.2 and Remark 11.1.2.1.2 hold:

1. For all diagrams in $\mathcal{C}$ of the form
   

   if we have

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

   then $\alpha =\beta $.

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