the proof of Theorem \[Onetoposmany\_loccon\], by using Theorem \[Onetoposmany\_cohcrit\].

\[Onetoposmany\_weaker\] *From the proof of the theorems it is clear that, by using the notion of dense subcategory, one can obtain alternative (although equivalent) versions of the criteria. For example, a weaker (in the ‘if’ direction) version of the criterion of Theorem \[Onetoposmany\_loccon\] reads as follows: a geometric theory $\mathbb T$ is classified by a locally connected topos (resp. connected and locally connected topos) if and only if there exists a collection $\cal F$ (resp. a collection $\cal F$ containing $\{[].\top\}$) of $\mathbb T$-indecomposable geometric formulae-in-context over $\Sigma$ such that for any geometric formula $\{\vec{y}. \psi\}$ over $\Sigma$, there exist objects $\{\vec{x_{i}}. \phi_{i}\}$ in $\cal F$ as $i$ varies in $I$ and $\mathbb T$-provably functional geometric formulae $\{\vec{x_{i}}, \vec{y}.\theta_{i}\}$ from $\{\vec{x_{i}}. \phi_{i}\}$ to $\{\vec{y}. \psi\}$ such that $\psi \vdash_{\vec{y}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}}(\exists \vec{x_{i}})\theta_{i}$ is provable in $\mathbb T$. Naturally, the ‘coherent’ version of this criterion also holds.*

We note that, by Theorem \[Onetoposmany\_cohcrit\], if $\mathbb T$ is coherent and $\phi(\vec{x})$ is a coherent formula over $\Sigma$ then $\phi(\vec{x})$ is $\mathbb T$-indecomposable if and only if for any finite family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same 