Reversed dual chain complex ($X_i={\mathrm{Hom}}_R(C_{-i},R),d_X\varphi =\varphi d_C$)

The reversed dual module of a chain complex becomes a chain complex with the unsigned precomposition differential above. A functional on $C_{-i}$ is sent to a functional on $C_{1-i}$, lowering the reversed degree by one.