If functions are defined as sets of ordered pairs, then how does it make sense to talk about the codomain of a function, or talk about a function being surjective (rather than saying something like “surjective to $X$”)?
If the codomain is actually part of the function, then does that mean that if $f : \mathbb{N} \to \mathbb{N}$ and $g : \mathbb{N} \to \mathbb{Z}$ such that $f(n) = g(n) = n$, then $f \neq g$?