In mathematics, especially category theory, a double category is a generalization of a category where instead of morphisms, we have vertical morphisms, horizontal morphisms and 2-morphisms. Introduced by Ehresmann in 1960s, the notion may be compared with that of a bicategory; namely, the notion of a bicategory is obtained by enrichment, while the notion of a double category is obtained by internalization. Precisely, a double category is a category internal to Cat (roughly meaning a category object).
Just as iterating the process of obtaining the notion of a 2-category leads to that of an n-category, iterating the process for a double category leads to that of an n-fold category.