When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair $(S,f)$?
I'm asking because it makes a difference to whether I need choice in some proofs or whether I don't. For example, if we prove that a denumerable union of denumerable sets is denumerable we need countable choice to prove it if we assume the former definition and we do not need choice at all if we assume the latter.