In How to get a group from a semigroup Arturo Madigin writes:
So if we look at the composition of the forgetful functors $\bf Group \to \bf Monoid \to \bf Semigroup$, we obtain a right adjoint by composing the adjoints going the other way, $\bf Semigroup \to \bf Monoid$ (adjoin a 1), and $\bf Monoid \to \bf Group$ (enveloping group). So: first adjoin a 1, then construct the enveloping group.
What are the properties of the composed functor: $F: \bf Group \to \bf Semigroup$? In particular is it full and/or faithful? Or are there multiple inclusion functors from which a particular one needs to be specified?
Faithful functor: $\forall (f,g: A \to B), Ff = Fg \implies f = g$
Full: $\forall (h: FA\to FB), \exists (f: A \to B):Ff = h$
As an FYI, as magma cites from Adamek & Herrlich's Abstract and Contrete Categories, a fully faithful functor reflects isos, sections, and retracts. (I assume it also preserves these properties). That may make it easier to think about it.
The context for this question is to compare and contrast the embedding of groups into semigroups versus metric spaces in topological spaces.