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