Can the inverse of set-valued function also be a set-valued function? Can the inverse of set-valued function also be a set-valued function? If so, what we call the set-valued function whose inverse is a single-valued function? Thanks very much!

Yes. A set-valued function is a function whose values are sets. The inverse of a set-valued function has to have an input that is a set. A function whose input is a set is called a set function. Therefore a set-valued function has a set-valued inverse iff the function is a set-valued set function. The inverse of such a function is also a set-valued set function.

You could say, the set-valued function whose inverse is a single-valued function is a single-value-argumented set-valued function, but you should declare that.