(1) Extend the codomain of f and g, to N, i.e. think of them as functions N -> N instead of functions N -> {0, ..., 9}.
(2) Compute the convolution of those two functions, giving you another (still finitely supported) function h: N -> N. Usually at this point h will have values that are larger than 10.
(3) Do all the carrying (e.g. repeatedly take the first n for which h(n) > 10, and subtract 10 from h(n) and add 1 to h(n+1)). Now all the values of h are in {0, ..., 9}, so you can think of h as a function N -> {0, ..., 9}.
The resulting function h is the base-10 representation of the product of the two numbers that f and g represent.
If being annoying by (not-)answering with "mu" to all gender/pronoun questions communicates it better, maybe it's a point in favor of being annoying!
IMO it points at some cracks in the perspective that there are different versions in the first place.