I'm interested in the following question.
Given a polynomial functor $F:Set \to Set$, does an initial F-algebra always exists?
It seems it is true. Reference:
The existence of initial algebras, and hence solutions of polynomial fixpoint equations over sets, is classical insight in Category Theory, going back at least to Lambek’s 1968 paper.
However, how hard is to show this? Do I need to read Lambek's paper or is there a simple proof? Are there any generalizations of the result I should be aware of?
Update
Here is a result coming from the reference int he comments:
Suppose $C$ is an $\omega$-category with binary products and coproducts, an initial element, and cartesian closed. Then polynomial endofunctors of $C$ have initial algebras.
It remains to show if $Set$ satisfies the above properties and to see if there is an algorithm building such algebras.