  We study lax epimorphisms in 2-categories, with special attention to
$\mathsf{Cat}$ and $\mathcal{V}$-$\mathsf{Cat}$. We show that any 2-category
with convenient colimits has an orthogonal $LaxEpi$-factorization system, and
we give a concrete description of this factorization in $\mathsf{Cat}$.

Category Theory