# Is there a category similar to Set, but having the morphism binary relations instead of functions?

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?

• I just want to add the note that while in one sense there are more morphisms in $\mathbf{Rel}$ than $\mathbf{Set}$, it's also the case that $\mathbf{Rel}$ is equivalent to what's called the Kliesli category on the covariant powerset monad. What that means is that all the morphisms in $\mathbf{Rel}$ can be viewed as a subclass of functions in $\mathbf{Set}$ equipped with a different composition operation. – Malice Vidrine Sep 13 '18 at 7:32