Coproduct of Free Monoids

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

Let $\map M A$ and $\map M B$ 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 $\map M {A \sqcup B}$ on $A \sqcup B$ is the coproduct of $\map M A$ and $\map M B$ in $\mathbf {Mon}$.

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

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


 * $\begin {xy}

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

<-5em,0em>*+{\map M A}        = "MA", <0em,0em>*+{\map M {A \sqcup B} } = "MAB", <5em,0em>*+{\map M B}         = "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 \mathop \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):


 * $j_1 = \overline {\paren {i_{A \mathop \sqcup B} \circ i_1} }$
 * $j_2 = \overline {\paren {i_{A \mathop \sqcup B} \circ i_2}}$

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