any family $\{\psi_{i} \textrm{ | } i\in I\}$ of geometric sentences, $\phi \vdash \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I'}} \psi_{i}$ provable in $\mathbb T$ for some finite subset $I'$ of $I$.

We note that any cartesian (resp. regular, coherent) theory $\mathbb T$ over a signature $\Sigma$ can be regarded as a geometric theory, and hence we have an equivalence of classifying toposes $[{{\cal C}_{\mathbb T}^{\textrm{cart}}}^{\textrm{op}}, {{\bf Set }}]\simeq {{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ (resp. ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{reg}}, J^{\textrm{reg}}_{\mathbb T}) \simeq {{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$, ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{coh}}, J^{\textrm{coh}}_{\mathbb T}) \simeq {{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$); moreover, it is immediate to see that for any cartesian (resp. regular, coherent) formula $\phi(\vec{x})$ over $\Sigma$, $y^{\textrm{cart}}_{\mathbb T}(\{\vec{x}. \phi\})$ (resp. $y^{\textrm{reg}}_{\mathbb T}(\{\vec{x}. \phi\})$, $y^{\textrm{coh}}_{\mathbb T}(\{\vec{x}. \phi\})$) corresponds under the equivalence to $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$.

This remark, combined with Propositions \[Onetoposmany\_identif\] and \[Onetoposmany\_express\], leads to the following results. Below, given a geometric theory $\mathbb T$ over a signature $\Sigma$, by saying that $\mathbb T$ is equivalent to a cartesian (resp. regular, coherent) theory we mean that $\mathbb T$ can be axiomatized by cartesian (resp. regular, coherent) sequents over $\Sigma$.

\[Onetoposmany\_cartcrit\] Let 