A homomorphism which is also a BiJection is called an isomorphism; two isomorphic objects are completely indistinguishable as far as the structure in question is concerned. A homomorphism from a set to itself is called an endomorphism, and if it is also an isomorphism is called an automorphism.
Any homomorphism f: X -> Y defines an EquivalenceRelation on X by a~b iff f(a)=f(b). The quotient X/~ can then be given an object-structure in a natural way, e.g. [x]*[y]=[x*y]. In that case the image of X is necessarily isomorphic to X/~. Note in some cases (e.g. GrouPs) a single equivalence class Y suffices to specify the structure of the quotient, so we write it X/Y.