Given a map $f : X \to Y$, the mapping cone $C(f)$ is the space obtained from the mapping cylinder $M(f)$ by identifying the subspace $X \times \{0\}$ to a single point.
How can I construct an isomorphism between the homology group $H_n(M(f),X \times \{0\})$ and the reduced homology group $\tilde{H}_n(C(f))$. I can prove they are isomorphic, but how can I construct actually the isomorphism between them? Any help please.