In detail, $f$ is representably full if, 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
\[ \beta =\text{id}_{f}\mathbin {\star }\alpha . \]