Indeed. The question "given a function as a set of ordered pairs, write code to decide if it is injective" has an answer, but replace that with "surjective" and suddenly you need to drag the codomain along as an extra parameter or the question makes no sense. This asymmetry bothers me.

EDIT: to be clear, "code" only works if your set is finite, but the principle of "does this question even have an answer" remains "yes" for injective and "no" for surjective without the codomain being named explicitly.