Sorry for my harsh comment; here's what I'm thinking of...
In math, you are dealing with many different kinds of object, not just numbers. In fact, one of the big realizations that led to modern mathematics is that not all mathematical objects can even be coded as numbers!
Your remark about "arrow" being more general than function is correct; but functions do not map only between sets of numbers, but between arbitrary sets, some of which contain elements that are not numbers, or even encodable as such.
