I call this hypotetic category BigSet. The definition of BigSet is:
- Objects are sets
- Morphisms are binary relations (including functions)
- Composition is the composition of relations
- Identities are the identity functions
Functions are special binary relations, so this category differs from Set only that in BigSet there are more morphisms than in Set. Am I right that this is a category? Is it used in mathematics? I yes, where, if not, why?