Left adjoint of an embedding

Question

Let R be be an embedding functor from C to D, that is R is injective-on-objects and faithful, and let L be its left adjoint, is there some property of L, and a good name for it, which R being an embedding induces via the adjunction?

Answer 1

It implies that any choice of counit $\epsilon$ is epimorphic. For arrows $f,g:a\to b$, $f\epsilon=g\epsilon$ iff $\epsilon LR(f)=\epsilon LR(g)$ iff $R(f)=R(g)$ and as $R$ is faithful that’s iff $f=g$. In fact we see $R$ is faithful iff the counit is always an epimorphism. But note we didn’t need injectivity on objects.