The codomain of a function f:A → B is {x : there is some a in A such that f(a) = x}, or, in Eindhoven notation, (∪ : a ∈ A : {f(a)}).
If the codomain of a function f:A → B is C, and B = C (i.e. if the range is equal to the codomain), then f is surjective. If B ≠ C, then there is a function g:A → C such that g(x) = f(x), and g is surjective.
Compare with domain and range.