I posted about monoid coproduct to try to understand the correct way for constructing them. In the process and from the answer I got, I am further confused whether the product and coproduct of plain monoids compare free monoids, what are the subtle differences in terms of what they look like. I understand in both coproduct of monoids and free monoids, the process of word reduction to strings are applied. I don't know anything beyond that. I am hoping someone can explain to me their subtle differences.
Thank you in advance.
If two monoids $M,N$ are free, say on sets $A$ and $B$, their coproduct $M*N$ is free as well, on the generating set $A+B$. You can prove this by hand, meaning you can prove that in $FA*FB$ there is no nontrivial relation between the elements, and that $A+B$ generates the monoid.
Or you can prove the universal property: functoriality of the free monoid construction gives monoid homomorphisms $FA\to F(A+B)\leftarrow FB$ induced by the inclusions $A\to A+B\leftarrow B$, whence a unique map $FA*FB\to F(A+B)$ by the universal property of coproducts. Now, this is a monoid isomorphism. (Alternatively: $FA\to F(A+B)\leftarrow FB$ is a coproduct cocone, which will imply $F(A+B)\cong FA*FB$ by uniqueness up to iso of a coproduct object.)
Or you can resort to slighty more abstract reasoning: $F : Set \to Mon$ is a left adjoint, hence it preserves colimits, hence coproducts: $F(A+B)\cong FA * FB$.