Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying \(\textbf{A}\) with \(\textbf{B}\) when there exists a strictly structure-preserving map \(\textbf{A}\rightarrow \textbf{B}\) that is genuinely (not just essentially) surjective in each dimension and faithful in the top dimension. We prove this principle for four types of structure: categories, monoidal categories, bicategories and double categories. The last of these theorems suggests that the right notion of equivalence between double categories is Campbell’s gregarious double equivalence, a conclusion also reached for different reasons in recent work of Moser, Sarazola and Verdugo.
