In my previous blog post I used, without proof, the fact that the initial algebra of the functor I + h \otimes - is a free monoid. The proof of this statement is not at all trivial and, frankly, I would never have been able to work it out by myself.

I worked my way through this proof, filling some of the steps that might have been obvious to a mathematician, but not to an outsider. I even learned how to draw diagrams using the TikZ package for LaTeX.

Part two: https://bartoszmilewski.com/2018/07/30/free-monoid-from-free-algebra-part-2/