context such that for each $i$, $\psi_{i}$ $\mathbb T$-provably implies $\phi$ and for any distinct $i, j\in I$, $\psi_{i}\wedge \psi_{j} \vdash_{\vec{x}} \bot$ is provable in $\mathbb T$, we have that $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash_{\vec{x}} \psi_{i}$ provable in $\mathbb T$ for some $i\in I$.

Regarding regular theories, their classifying toposes are always connected and locally connected; this was already observed in [@El2], and also follows from our Remark \[Onetoposmany\_Rem\].

We have already noticed that any atomic topos $\cal E$ is locally connected; in fact, as it is observed in [@El2], the atoms of $\cal E$ are exactly the connected (equivalently, indecomposable) objects of $\cal E$. Hence, in view of Theorem \[Onetoposmany\_car\], by replacing ‘$\mathbb T$-indecomposable’ with ‘$\mathbb T$-complete’ everywhere in the criteria above, one obtains syntactic criteria for a geometric (resp. coherent) theory to be classified by an atomic topos; also, one can obtain alternative versions of these criteria in the same spirit as in Remark \[Onetoposmany\_weaker\].

Concerning atomic toposes, let us notice that if $\phi(\vec{x})$ is a $\mathbb T$-complete formula then $\phi(\vec{x})$ is $\mathbb T$-provably equivalent to a regular formula; indeed, this follows immediately from the fact that any geometric formula 