Coproduct of Free Monoids

Theorem
Let $\mathbf{Mon}$ be the category of monoids.

Let $M \left({A}\right)$ and $M \left({B}\right)$ be free monoids on sets $A$ and $B$, respectively.

Let $A \sqcup B$ be the disjoint union of $A$ and $B$.

Then the free monoid $M \left({A \sqcup B}\right)$ on $A \sqcup B$ is the coproduct of $M \left({A}\right)$ and $M \left({B}\right)$ in $\mathbf{Mon}$.

Proof
By Coproduct is Unique, it suffices to verify that $M \left({A \sqcup B}\right)$ is a coproduct for $M \left({A}\right)$ and $M \left({B}\right)$.

By the UMP of $M \left({A}\right)$, $M \left({B}\right)$ and $M \left({A \sqcup B}\right)$, we have the following commutative diagram:


 * $\begin{xy}

<0em, 5em>*+{N} = "N",

<-5em,0em>*+{M \left({A}\right)}        = "MA", <0em,0em>*+{M \left({A \sqcup B}\right)} = "MAB", <5em,0em>*+{M \left({B}\right)}         = "MB",

<-5em,-5em>*+{A}        = "A", <0em,-5em>*+{A \sqcup B} = "AB", <5em,-5em>*+{B}         = "B",

"A";"MA" **@{-} ?>*@{>}  ?*!/_.8em/{i_A}, "B";"MB" **@{-} ?>*@{>}  ?*!/^.8em/{i_B}, "AB";"MAB" **@{-} ?>*@{>} ?*!/_.8em/{i_{A \sqcup B}},

"A";"AB" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, "B";"AB" **@{-} ?>*@{>} ?*!/_.8em/{i_2},

"MA";"MAB" **@{-} ?>*@{>} ?*!/_.8em/{j_1}, "MB";"MAB" **@{-} ?>*@{>} ?*!/^.8em/{j_2},

"MA";"N" **@{-} ?>*@{>}  ?*!/_.8em/{\bar f}, "MB";"N" **@{-} ?>*@{>}  ?*!/^.8em/{\bar g}, "MAB";"N" **@{--} ?>*@{>} ?*!/_.8em/{\bar h}, \end{xy}$

Here (in the notation for free monoids):


 * $j_1 = \overline {\left({i_{A \sqcup B} \circ i_1}\right)}$
 * $j_2 = \overline {\left({i_{A \sqcup B} \circ i_2}\right)}$

and $i_1$, $i_2$ are the injections for the coproduct.