Question about mapping cones in homology theory

Please see the image attached for the question I would post the question in the text box but I don't know how to render all the arrows and diagrams manually, hence the picture

The "homology theory" in question simply refers to an abstract theory satisfying the Eilenberg-Steenrod axioms

As you have constructed it, mapping cone $C_f$ fits into $X\longrightarrow Y \longrightarrow C_f \longrightarrow SX\longrightarrow SY$ where $S$ is the suspension and the map $C_f\to SX$ collapses the copy of $Y$ in $C_f$. What is important is the segment

$$Y\longrightarrow C_f \longrightarrow SX.$$

One of the axioms of a generalised homology theory is that the connecting morphism $H_n(SX) = H_{n-1}(X) \longrightarrow H_{n-1}(Y)$ identifies with $f$ in the LES. This allows you to conclude. If you are working with usual singular homology, maybe it is something you want to prove.

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