# initial algebra and free monoid

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

then by going through the $-\otimes A \dashv -^A$ adjunction I get and algebra for $A^{*A^*}$ so I get an arrow from $A^*$ by initiality

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

and that

But I am not sure how to prove that the universal property is verified :

Any reference or tips appreciated

The solution given in diagram

In the right upper part of the diagram, you can reduce $F(\mathsf{free} f) \circ \mathsf{inr}$ to $\mathsf{id} \otimes \mathsf{free} f : A \otimes A^{*} \to A \otimes G$.
You end up with $(\mathsf{id} \otimes \mathsf{free} f) \circ (\mathsf{id} \otimes e)$, which is $\mathsf{id} \otimes e_G$. There you can swap the order with $f \otimes \mathsf{id}$ (a.k.a. $- \otimes -$ functoriality) to apply $(G, m_G, e_G)$'s right identity law.
For the uniqueness part, I think you can use the initiality of $A^{*}$.