Cellular chain complex of a mapping cone (${\tilde{C}}_q(C_f)=C_q(Y)\oplus C_{q-1}(X)$)

For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differentialHere $C_{-1}(X)=0$ and the degree-zero group identifies a vertex $y$ with the reduced chain $y-v$, where $v$ is the cone vertex. Thus this is the mapping cone of $f_{\#}$, in the displayed ordering of the summands.