If an abelian category $C$ is small, then the category $Ind(C)$ is a Grothendieck category. Now suppose $C$ is any small abelian category. Then $Pro(C)$, the category of all cofiltered limits of objects in $C$ should be an abelian category. However, I see no reason why it should be small (we could take limits over categories of any cardinality).
Is it still possible for $Ind(Pro(C))$, the category of filtered colimits in $Pro(C)$, to be a Grothendieck category?