 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 $1\to 0$. The category $\mathcal{C}$ is not balanced.