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Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is acomplete DVR, we first prove a theorem of meromorphic descent along a possiblyinfinite cover of $\hat{X}$. Using this we construct a specialization functorfrom the category of continuous representations of the pro-\'etale fundamentalgroup of the special fiber to the category of $F$-divided sheaves on thegeneric fiber. This specialization functor partially recovers thespecialization functor of the \'etale fundamental groups. We also express thepro-\'etale fundamental group of a connected scheme $X$ of finite type over afield as coproducts and quotients of the free group and the \'etale fundamentalgroups of the normalizations of the irreducible components of $X$ and those ofits singular loci.

A theorem on meromorphic descent and the specialization of the pro-\'etale fundamental group