Well, I don't find any relevance in the cohomology in your question. The notation for n-th homology groups is , while the notation for n-th cohomology group is .

Is your question saying

"If an isomorphic chain map is given between two chain complexes, how do we prove that homology groups of two chain complexes are isomorphic?"

If so, define an isomorphic chain map as , and first use commutative diagram to show , where is a boundary map. Then, deduce that homology groups of two chain complexes are isomorphic.