I have recently been advised, in a particular example which is described below, that taking quotients of a category can be complicated, but I don't see where exactly lies the difficulty.
More precisely, I wanted to build a category as such: I take a group-as-category $G$ (i.e. a category with a single object $A$ and invertible morphisms), and form the free strict monoidal category generated by $G$ and a morphism $\mu \colon A^{\otimes 2} \to A$. Then I quotient the category by the relation $\mu \circ k = (k \otimes k) \circ \mu$, for all $k \in G$.
I would thus obtain the free strict monoidal category generated by $G$, $\mu \colon A^{\otimes 2} \to A$, and the above relation, wherein the morphisms are binary forests whose leaves are labelled by elements of $G$.
Is there something wrong with the construction of this category ?