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Let $\mathbb{D}$ be the category of pro-sets (or abelian pro-groups). It isproved that for any Grothendieck site $X$, there exists a reflector from thecategory of precosheaves on $X$ with values in $\mathbb{D}$ to the fullsubcategory of cosheaves. In the case of precosheaves on topological spaces, itis proved that any precosheaf is smooth, i.e. is locally isomorphic to acosheaf. Constant cosheaves are constructed, and there are establishedconnections with shape theory.

Precosheaves of pro-sets and abelian pro-groups are smooth