A W-type is an initial algebra for a polynomial endofunctor $P$ on a category $C$. A well-known example is that of a natural numbers object (NNO). Usually it is assumed that $C$ is locally cartesian closed (lcc), but this only ensures that all polynomial endofunctors exist so as to ask whether the corresponding W-type exists, and it also simplifies the definition of W-type (one needs to use parameters if not in the lcc case, see for example the definition of an NNO in a merely finite-product category). One can always, in a lextensive category, talk about *finitary* polynomial endofunctors: these are 'really' polynomials, namely endofunctors of the form
$$
X \mapsto \coprod_{i \leq n} A_i \times X^{\times i},
$$
for some collection of objects $\lbrace A_i\rbrace_{i \leq n}$, or, if countably extensive, with a coproduct over $\mathbb{N}$ (the *external* natural numbers), and this is the case in the example I'm particularly interested in (and I know it is *not* lcc, just a cocomplete pretopos with NNO, in the appropriate sense).

My question is this:

If I have an infinitary lextensive category (so

allcoproducts, compatible with pullbacks) that is locally cartesian closed, with an NNO, what W-types do I get for free? What if the category is not lcc?

The first question at least puts an upper bound on what I can expect, the second one is closer to what I expect to see.

well-powered. Of course, the well-poweredness assumption might be far too drastic for your purposes. $\endgroup$