isomorphisms external direct product

Suppose we have some isomorphism φ which maps Z(subscript 3) + Z(subscript 5)

(ie the external direct product between Z3 and Z5..sorry dont know how to make the symbols) to Z(subscript 15)

If φ(2,3) = 2 which element in the external direct product maps to 1 in Z15

This is screaming chinese remainder thm to me but it doesnt work. I thought for sure the mapping would be something along the lines of 2*3 + 3*5 = ___ (mod 15)

but clearly doesnt work for the given example. ive tried numerous other different combinations of the same type of idea I just feel like im playing a "guess and check" type of game, any thoughts on a method?