I'm trying to understand a free groupoid on a directed chain of n vertices. I know the objects will be the vertices of our graph and the morphisms will be concatenations of the edges in the graph and formal inverses to them. But when studying the free category of a directed chain or any graph, we required that to create morphisms we must have the source of one arrow equal the range of another, but is this needed in the free groupoid?
So if I have the free groupoid of a directed chain of n vertices, we have n objects (1 for evcery vertex), $n$ identiy morphisms and morphisms $f_1: v_1 \to v_2$ sending vertex 1 to vertex 2 and $f_2: v_2 \to v_3$, $f_3, ..., f_{n-1}$ and their inverses $f_1^{-1}: v_2 \to v_1$ and $f_2^{-1},..., f_{n-1}^{-1}$ but do we have a morphism such $f_1 \circ f_3$, the definiton only mentioned concatenation so I'm not exactly sure if we require the range and domain to match up.
If this is the case then we can never concatenate something like $f_{1}^{-1} \circ f_2 \circ f_5 \circ f_6^{-1}$?