The very concept of functors was created to describe sets that did exactly that. The functor laws only require that there is an operator that can wrap a value f

The very concept of functors was created to describe sets that did exactly that. The functor laws only require that there is an operator that can wrap a value from one category into the functor's target category, that mapping with the identity function is the same as the identity function, and that the results of composition of mapping with two functions is the same as the results of mapping with the composition of the two functions. Set fulfills these laws perfectly. There is no law for maintaining cardinality, because the functor laws don't care if cardinality is a concept that even makes sense for the target category in question.