Internal category

In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal categories consist of a pair of objects in the ambient category—thought of as the 'object of objects' and 'object of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities. Group objects, are common examples of internal categories.
There are notions of internal functors and natural transformations that make the collection of internal categories in a fixed category into a 2-category.
Definitions.
Let formula_1 be a category with pullbacks. An internal category in formula_2 consists of the following data: two formula_2-objects formula_4 named "object of objects" and "object of morphisms" respectively and four formula_2-arrows formula_6 subject to coherence conditions expressing the axioms of category theory. See 