Cosheafification

It is proved that for any Grothendieck site $X$, there exists a coreflection (called $\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{e}\mathbf{a}\mathbf{f}\mathbf{i}\mathbf{f}\mathbf{i}\mathbf{c}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}$) from the category of precosheaves on $X$ with values in a category $\mathbf{K}$, to the full subcategory of cosheaves, provided either $\mathbf{K}$ or ${\mathbf{K}}^{op}$ is locally presentable. If $\mathbf{K}$ is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category $$\mathbf{P}\mathbf{r}\mathbf{o}$% \left( $\mathbf{K}$\right) $ofpro-objectsin$$\mathbf{K}$$. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in $\mathbf{P}\mathbf{r}\mathbf{o}\left(\mathbf{K}\right)$ is $\mathbf{s}\mathbf{m}\mathbf{o}\mathbf{o}\mathbf{t}\mathbf{h}$, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.