The Clowder Project

In detail, $f$ is representably fully faithful if the conditions in Remark 13.1.1.1.2 and Remark 13.1.2.1.2 hold:

1. 1.

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

   
   if we have
   \[ \operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\beta , \]

   then $\alpha =\beta $.

2. 2.

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