what you are being asked to do is FIND a definition for * so that φ(ab) = φ(a)*φ(b).

the fact that φ is 1-1 and onto already is the "iso" part....ensuring that φ(ab) = φ(a)*φ(b) is the "morphism" part.

suppose that a*b is defined as ab+1...does this definition of * fit the requirements?

(your question is a little vague...since no other information is given, i am assuming you aren't requiring that * be associative, or have an identity).