Is there a left adjoint $F$ to the "forgetful" inclusion functor $U$ from the category of groups (interpreted as groupoids with one object $*$) to the category of groupoids? If so, then letting $\eta$ be the unit of the adjunction, is $\eta_G : G \to UF(G)$ faithful for every groupoid $G$?
I suspect this question is related to the construction of free products of groups, and in the case that a groupoid is a disjoint union of groups, then $F$ specializes to the free product of the groups (indeed, if a left adjoint exists then it preserves coproducts). My intuition says we could take words on all the morphisms regardless of composability, and then allow composable morphisms to reduce when they are adjacent in a word. If this intuition is correct, the question I am most interested in is whether this "free group on a groupoid $G$" has an injective inclusion from each $G(x, y)$, analogous to how the free product of groups has injective inclusions from each factor.
What about the corresponding situation with categories of monoids and categories?
Let us first suppose $G$ is a connected groupoid; fix an object $x$ of $G$. For each other object $y$ of $G$, also fix an isomorphism $f_y:x\to y$. If $H$ is a group, then a functor $G\to H$ is uniquely determined by a homomorphism $\operatorname{Aut}(x)\to H$ together with a choice of element of $H$ to map each $f_y$ to. It follows that the left adjoint $F(G)$ evaluated on $G$ can be described as the free product of $\operatorname{Aut}(x)$ with a free group on the set of objects of $G$ other than $x$.
For a general groupoid $G$, then, $F(G)$ is the free product of the groups given by each of its connected components in this way. In particular, then, since the inclusions of groups into a free product of groups are injective, the unit of the adjunction $\eta_G:G\to UF(G)$ is faithful.
Here is an alternative description of the left adjoint that also works just as well for the forgetful functor from monoids to categories. An element of $F(G)$ is a finite sequence of non-identity morphisms of $G$ such that no two consecutive morphisms in the sequence are composable (call such sequences reduced). Composition of two such finite sequences is defined by concatenation, and then turning it into a reduced sequence by repeatedly composing any adjacent composable morphisms in the sequence and removing any identity morphisms. (Some work is then needed to check that this is associative; this is very similar to checking associativity of the multiplication in the free product of groups constructed using words.) This description makes it even easier to see that the unit of the adjunction is faithful (and also that it is faithful in the case of categories and monoids).
By the way, if you just want existence of the left adjoints, that is easy by abstract nonsense. For instance, it follows from the adjoint functor theorem, since groups are closed under limits in the category of groupoids (or monoids are closed under limits in the category of categories). Or concretely, you can just take the set of strings representing formal compositions of morphisms of your groupoid (or category) and say that two strings are equivalent if they become equal under every functor to a group (or monoid). A description like this isn't very useful for figuring out whether the unit is faithful, though, unless you can get a better handle on that equivalence relation.