I am missing something to prove that the initial algebra $A^*=\mu x. Fx$ of the functor $FX=I+A \otimes X $ is the free monoid in a monoidal category.
Here's one start
Summing up I can build 2 arrows in blue like so
So I get $m$ coming back through the adjunction and $e= \delta . inl$ as given. So, I have a monoid $(A^*,m,e)$, and I can put A things in it, that's a good start.
Now, take a monoid $(G,e,m)$, a function f from A to G, the paper claims the initial algebra verifies the universal property
But I am not sure how to prove that the universal property is verified :
Any reference or tips appreciated