On a generalized modularization theorem

The relation between a metalogical property of entailment (interpolation) and a structural property of categories of theory presentations (stability of faithful morphisms under pushouts) is studied in a general logical framework. The stability of faithful morphisms for any specific instance of the pushout construction is shown to be equivalent to the existence of interpolants over the corresponding locus of the underlying logic. The latter neither assumes nor requires interpolation to be a global property of the formalism.