Properties of Group $\to$ Monoid $\to$ Semigroup. 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. 
 A: The forgetful functor from groups to semigroups is fully faithful. This is equivalent to the statement that a group homomorphism $f : G \to H$ between two groups is the same thing as a semigroup homomorphism. To see this, first suppose that $f$ is a semigroup homomorphism. Then $f(e) = f(e^2) = f(e)^2$ (where $e$ is the identity in $G$), from which it follows by cancellation that $f(e)$ is the identity in $H$. Second, $f(e) = f(g^{-1} g) = f(g^{-1}) f(g)$, from which it follows by the uniqueness of inverses that $f(g^{-1}) = f(g)^{-1}$. 
(Note that it is not true that the forgetful functor from monoids to semigroups is full. The problem is that the identity of a monoid need not be sent to the identity under a semigroup homomorphism, but just to an idempotent.) 
The properties of the forgetful functor from metric spaces to topological spaces depends on what category of metric spaces you're using. A common choice is to take metric spaces and continuous functions, but this is the "wrong" notion of morphism for metric spaces (it's the correct notion of morphism for metrizable spaces); you can't recover the metric from an isomorphism class in this category. If you want to recover metrics from isomorphism classes, the "correct" notion of morphism between metric spaces is a weak contraction. 
