is an irreducible object of ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$;

\(iv) $\phi(\vec{x})$ is $\mathbb T$-compact if and only if $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is a compact object of ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$;

\(v) $\phi(\vec{x})$ is $\mathbb T$-supercompact if and only if $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is a supercompact object of ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$.

Parts (iii), (iv) and (v) follow immediately from Proposition \[Onetoposmany\_express\], by using Lemma D1.4.4(iv) [@El2] and the fact that cover-mono factorizations of arrow exist in ${\cal C}_{\mathbb T}^{\textrm{geom}}$.

Parts (i) and (ii) follow from the fact that in a topos coproducts are the same thing as disjoint unions, since $y^{\textrm{geom}}_{\mathbb T}({\cal C}_{\mathbb T}^{\textrm{geom}})$ is closed in ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ under taking subobjects and $y^{\textrm{geom}}_{\mathbb T}:{\cal C}^{\textrm{geom}}_{\mathbb T}\to {{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$ is a full and faithful geometric functor.

Recall from [@El2] that a Grothendieck topos $\cal E$ is compact if and only if the terminal object of $\cal E$ is compact; hence, a geometric theory $\mathbb T$ over a signature $\Sigma$ is classified by a compact topos if and only if $y^{\textrm{geom}}_{\mathbb T}(\{[]. \top\})$ is compact in ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$, if and only if for 