Mapping cone ($M(f)$)

For a chain map $f:C\to C^{\prime }$, one mapping-cone convention takes $M(f{)}_i=C_{i-1}\oplus C_i^{\prime }$ with a differential combining $d,d^{\prime }$ and $f$. Its short exact sequence with $C^{\prime }$ and a shift of $C$ produces a long exact sequence in homology.