Prove that the rule $f([a]_{mn}) = ([a]_m,[a]_n)$ determines a well-defined function $f : Z/mnZ \to Z/mZ × Z/nZ.$

Question

I think it is well defined because of the fact that it will always produce a unique output. Even if [a]_2 repeats as the first term where mn is even, [a]_n will be unique so the output itself will be unique. Is this thinking correct?