Pointwise-monic adjunction over a non-balanced poset
= Pointwise-monic adjunction over a non-balanced poset
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 $1\to0$. The category $\mathcal C$ is not balanced.