The Clowder Project

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

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

   if we have

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

   then $\alpha =\beta $.

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