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Let $C$ be a model category given by generators and relations in the sense of Dugger (that is, $C$ is a left Bousfield localization of a global projective model model structure on simplicial presheaves on a small category $K$).

Does $C$ have a fibrate replacement functor that preserves finite limits?

As far as I understand, in the case when $C$ is just the category of simplicial presheaves, then the functor $[K^\mathrm{op}, \mathrm{Ex}^{\infty}] \colon[K^\mathrm{op}, \mathrm{sSet}] \to [K^\mathrm{op}, \mathrm{sSet}]$ is the desired one (and it also has other remarkable properties).

But in the case of localization, I have so far only found a mention that there is something about ∞-stackification in HTT 6.5.3 and viewing it has not helped me yet.

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    $\begingroup$ Is there any reason to expect this to be true? (There is one obstruction I can think of: if finitary products are not homotopical then there cannot even be a fibrant replacement functor that preserves finitary products.) $\endgroup$
    – Zhen Lin
    Commented May 26, 2023 at 2:38
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    $\begingroup$ In the case of ∞-toposes modeled by the local projective model structure, the existence of such a fibrant replacement functor can be deduced from the explicit construction of the associated ∞-sheaf given by the Verdier hypercovering theorem. $\endgroup$
    – Dmitri Pavlov
    Commented May 26, 2023 at 9:32

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