A functor $F : C \to D$ between coherent categories is called coherent if it is a regular functor and in addition preserves finite unions.
For $\mathcal{T}$ a coherent theory and $\mathcal{C}_{\mathbb{T}}$ its syntactic category, coherent functors $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ into some topos $\mathcal{E}$ are precisely models of the theory in $\mathcal{E}$.
For $\mathcal{C}_{\mathbb{T}}$ equipped with the structure of the syntactic site (the coherent coverage), this is in turn equivalent to geometric morphisms $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the sheaf topos over $\mathcal{C}_{\mathbb{T}}$ (the classifying topos for the theory).
coherent functor
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