I was wondering what is the precise relation between initial $F$-algebras and free $F$-algebras. These concepts seem very similar, as initial algebras can be used to define recursive constructions over a functor $F$, and similarly free algebras can be used to construct expression trees.
Just to give more context, I was trying to code a Free Monad over a functor F, and I realized that my implementation worked using concepts of fix and unfix from initial algebras, as well as catamorphisms.
This discussion can be seen in this post. I didn't grasp everything said in the link, but from what I understood, given a functor $F$, consider, for example, the functor $Free$ that defines free monoids. Thus, $Free F$ would be the free monad over $F$. Now, the post then claims that $(Free F \varnothing, fix)$ would be the initial $F$-algebra.
The discussion ends with the assertion " any free object is initial in the corresponding comma category". Which I don't quite understand.