On the equivalence invariance of formal category theory

We prove a result of equivalence invariance of formal category theory for statements that can be expressed within an equipment. To do this, we exploit Henry and Bardomiano Martínez’s link between Makkai’s FOLDS (first order logic with dependent sorts) and abstract homotopy theory. In the process, we construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares, and whose fibrant objects are the equipments.