Given a free group $F_X$, and it's Cayley graph, $Cay(F_X,X)$, we can take the quotient graph of $Cay(F_X,X)$ by the action of $F_X$ on it and obtain $R_{|X|}$, a "bouquet of circles" or "rose" consisting of a single vertex with $|X|$ edges labeled by the generators of $F_X$.
When learning this, I was stricken by the similarity of the bouquet of circles with the categorical definition of a group: Namely, a group may be considered a category with a single object in which all morphisms are isomorphisms. If we draw the object of the category as a vertex and the morphisms as directed edges, we obtain a similar graph (although, clearly, there is a difference, since in the case of the category-based graph, edges are labeled by elements, while in the case of the rose, edges are labeled by generators).
Is there any relationship between these two constructions, or is this merely a coincidence?