Free monoids and Riguet congruences

For a set $A$, after recalling the notion of free monoid on $A$, denoted by $\mathbf{A}^{\star}$, we associate to $\mathbf{A}^{\star}$ a category $\mathsf{C}(\mathbf{A}^{\star})$, which, in general, is not skeletal, and prove that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$, the category of finite $A$-sorted sets. Following this, after recalling and completing the notion of Riguet congruence on a category, we obtain, for a suitable Riguet congruence on $\mathsf{C}(\mathbf{A}^{\star})$, a skeletal quotient category $\mathsf{Q}(\mathbf{A}^{\star})$ of $\mathsf{C}(\mathbf{A}^{\star})$ and prove that it is also equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$.