Pairing is caring /

Dear thinking people,

let N denote the natural numbers and let [n] denote a set with n elements.
Fix a natural number m.

I'm looking for a bijection between N x N, and the function space [3] --> [m] x [6].
In other word, I need a one-one mapping between the pairs of natural numbers and the key-value pairs with 3 different possible keys that take values in the set of pairs from [m] x [6].

It might be simpler to even relax the problem and replace [m] with N, although I'll only be really interested in functions that map into the first m integers in that pair.

On a related note, since I know there are bored strong programmers out there, how are such small function spaces best encoded in the language C#, for example? If anyone is up for a challenge, try coding up the back and forth of the related Cantor pairing function:
https://en.wikipedia.org/wiki/Pairing_function

Best Regards
qed