Pointwise-monic unit-and-counit criterion

Suppose $\mathcal{C}$ is 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 and its essential image is closed under strong quotients.

Let $\mathcal{C}=\{0<1\}$ and let $\mathcal{D}$ be the terminal category. The unique functor $F:\mathcal{C}\to \mathcal{D}$ is left adjoint to the functor selecting $1$. Every arrow in a poset is monic, so the unit and counit are pointwise monic, but $F$ is not full because the unique arrow $F1\to F0$ has no preimage $1\to 0$. The category $\mathcal{C}$ is not balanced.