nLab
free groupoid

Given a (directed) graph $D$, that is, a collection of vertices and of labeled arrows between them, the free groupoid $G(D)$ on$D$ is the category – in fact the groupoid – that has the vertices of $D$ as objects, and whose morphisms are constructed recursively by formal composition (i.e., juxtaposition) from identity maps, the arrows of $D$ and formal inverses for the arrows of $D$. The only relations between morphisms of $G(D)$ are the necessary ones defining the identity of each object, the inverse of each arrow in $D$ and the associativity of composition. This is clearly a groupoid, which comes with an evident morphism $D\to G(D)$ of directed graphs.

The above sketched construction could be made more precise, but what really matters is the universal property it enjoyes: the free groupoid $G(D)$ is the universal (initial) groupoid mapping out of $D$. By varying $D$, the free groupoid yields a functor $G$ from directed graphs to groupoids, left adjoint to the forgetful functor.

This last conceptual characterization is best taken as the definition. Similarly, it is possible to construct the left adjoint to the forgetful functor from groupoids to categories, that is the free groupoid over a category.

Related notions

A quiver category is the free category on a graph.