The Clowder Project

In detail, a $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is pseudomonic if it satisfies the following conditions:

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 $2$-isomorphism

   
   of $\mathcal{C}$, there exists a $2$-isomorphism
   
   of $\mathcal{C}$ such that we have an equality
   
   of pasting diagrams in $\mathcal{C}$, i.e. such that we have
   \[ \beta =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha . \]