Let $\mathcal A$ be the category of Arens-Michael algebras, that is, projective limits of Banach algebras. Since $\mathcal A$ is a concrete category, an $\mathcal A$-valued presheaf $A$ admits a set-valued sheafification $A_{S.}$ I would like to know if there is a good way to associate an $\mathcal A$-valued sheaf to $A_{S.}$

Now, as I gather from nLab's article on sheafification and Kashiwara and Schapiro's *Categories and Sheafs,* if the category $\mathcal C$ is such that:

- Small projective and small inductive limits exist,
- Small filtrant limits are exact,
- The IPC property holds,

then $\mathcal C$-valued presheaves admit a $\mathcal C$-valued sheafification. Thus, my question is: does the category $\mathcal A$ of Arens-Michael algebras have these properties? And, in the negative case, is there a category of topological algebras, containing $\mathcal A$ as a subcategory, having these properties?