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Any small pretopos $C$ can be embedded into a Grothendieck topos by a fully faithful functor that preserves all the pretopos structure (limits, images, finite unions of subobjects, disjoint coproducts, and quotients of equivalence relations). Namely, we may consider the topos of sheaves for the coherent topology on $C$, with the sheafified Yoneda embedding. If $C$ is (locally) cartesian closed, then that structure is also preserved by this embedding.

My question is, what if $C$ also has a natural numbers object, or more general initial algebras for special endofunctors (e.g. "W-types")? Can we embed it into a topos of sheaves in a way that preserves these initial algebras?

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