then follows from Lemma \[Onetoposmany\_equiv\](ii) and the fact that the functor $y^{\textrm{geom}}_{\mathbb T}$ is geometric and full and faithful.

Conversely, if the condition of the criterion is satisfied then we have, by Lemma \[Onetoposmany\_equiv\](ii) and Proposition \[Onetoposmany\_denseness\](ii), that the objects of the form $y^{\textrm{geom}}_{\mathbb T}(\{\vec{y}. \psi\})$ for a $\mathbb T$-indecomposable formula $\psi(\vec{y})$ form a separating set for\
${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ made of indecomposable objects; then ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ is locally connected by Theorem \[Onetoposmany\_car\](i).

The following result is the coherent analogue of this theorem.

Let $\mathbb T$ be a coherent theory over a signature $\Sigma$. Then $\mathbb T$ is classified by a locally connected topos (resp. connected and locally connected topos) if and only if for any coherent formula $\phi(\vec{x})$ over $\Sigma$ there exists a (unique) finite family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of $\mathbb T$-indecomposable geometric formulae in the same context such that

\(i) for each $i$, $\psi_{i}$ $\mathbb T$-provably implies $\phi$,

\(ii) for any distinct $i, j\in I$, $\psi_{i}\wedge \psi_{j} \vdash_{\vec{x}} \bot$ is provable in $\mathbb T$, and

\(iii) $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ is provable in $\mathbb T$\
(resp. and $\{[].\top\}$ is $\mathbb T$-indecomposable).

The proof proceeds analogously to 