Pointwise-monic unit-and-counit criterion
= Pointwise-monic unit-and-counit criterion
Suppose $\mathcal C$ is <balanced category>[balanced] and every morphism of $\mathcal D$ factors as a <strong epimorphism> followed by a <monomorphism>. For $F:\mathcal C\rightleftarrows\mathcal D:G$, both unit and counit are pointwise monic exactly when $F$ is <full and faithful functor>[full and faithful] and its essential image is closed under <strong quotient>[strong quotients].