I have $2$ pair of chain complexes $(X_1,Y_1), (X_2,Y_2)$. And $$H_i(X_1,\mathbb Z) \cong H_i(X_2,\mathbb Z), H_i(Y_1,\mathbb Z) \cong H_i(Y_2,\mathbb Z).$$ Is it true that $$H_i(X_1, Y_1,\mathbb Z) \cong H_i(X_2,Y_2, \mathbb Z) ?$$ The first idea is to use $5$-lemma for the exact sequence of of homology. But I can't find homomorphism $$f: H_i(X_1, Y_1,\mathbb Z) \to H_i(X_2,Y_2, \mathbb Z),$$ commuting with isomorphisms of $$H_i(X_1,\mathbb Z)\ and\ H_i(X_2,\mathbb Z),\ H_i(Y_1,\mathbb Z)\ and\ H_i(Y_2,\mathbb Z).$$
UPD It isn't true. $X_1=Y_1=M, M$ is Möbius strip, $X_2=M=X_1, Y_2=S^1$ - the bound of M.