Morita equivalence of geometric theories ($\mathbf{Set}[\mathbb{T}]\simeq \mathbf{Set}[\mathbb{S}]$)

Geometric theories are Morita-equivalent when their classifying toposes are equivalent. This gives equivalent internal model categories pseudonaturally in every Grothendieck topos. A common classifying topos transports intrinsic invariants between different presentations; agreement of set-valued model categories alone is insufficient. The duality between geometric quotients and subtoposes transfers geometric theory extensions along such an equivalence.