Polynomial functors preserving ω-colimits

Question

I'm working my way through Steve Awodey's Category Theory book and on p.271, Proposition 10.12 says:

If the category $S$ has an initial object $0$ and colimits of diagrams of type $\omega$ (call them "$\omega$-colimits"), and the functor $P: S \to S$ preserves $\omega$-colimits, then $P$ has an initial algebra.

Shortly thereafter it states polynomial functors $P: \mathbf{Sets} \to \mathbf{Sets}$ preserve $\omega$-colimits.

My intuition is telling me that because polynomial functors are of the form

$$P(X) = C_{0} + C_{1} \times X + C_{2} \times X^{2} + ... + C_{n} \times X^{n}$$

and

$$+ \dashv \Delta \dashv \times $$

$+$ preserves colimits wrt $\Delta$ (the $\Delta$ handling the case of $X^{n}$) which in turn preserves colimits wrt $\times$ (the coefficient case).. and then since $\times$ preserves products on $\mathbf{Sets}$ and $\mathbf{Sets}$ has $\omega$-colimits (I'm not sure about this one..), we're done.

Is that an approximate line of reasoning?

Answer 1

I don't know what you mean by "colimits wrt $\times$." You need to show three things:

1. that coproducts commute with colimits,
2. that product distributes over colimits, and
3. that $X \mapsto X^n$ preserves $\omega$-colimits.

You haven't addressed the third point, and it's also not true that $X \mapsto X^n$ commutes with arbitrary colimits, so this is in some sense the hardest step.