Left adjoint of diagonal functor

Question

How would i go about showing that the diagonal functor ∆ : Mon → Mon × Mon has a left adjoint?

Answer 1

The first thing you want to see is that such a left adjoint gives you binary coproducts. Let's denote the object to which the left adjoint takes $\langle A,B\rangle$ by $A\square B$. Then left adjointness tells us that there is a natural correspondence $$\dfrac{A\square B\longrightarrow C}{\langle A,B\rangle\longrightarrow\Delta(C)}.$$ That is, a morphism $f:A\square B\to C$ corresponds to a unique pair of arrows $f_1:A\to C$ and $f_2:B\to C$; which is just what a coproduct of $A,B$ is.

How the coproduct is actually constructed is a little more complicated. The easiest construction for $A+B$ is indeed as a set of strings (or rather, equivalence classes of strings), but not quite as you've suggested. What you use is strings of elements from $A\cup B$, quotiented by an equivalence relation that respects all the equalities in $A$ and $B$ (the details are hairy, but what you're looking for is a free product of monoids).