I'm trying to understand where the identity
$$ i^{n+1} \circ \pi ^{n+1} \circ d_B^{n} \circ s^n = d_B^n \circ s^n - s^{n+1} \circ d_C^n $$
comes from here. I've tried to make use of the commuting properties of some of the maps involved, and the identities $q^n \circ s^n = \text{id}_{C^n}$ and $\pi^n \circ i^n = \text{id}_{A^n}$, but haven't got anywhere. In particular, I can't see how we've managed to get a difference of two maps on the RHS. Can anyone shed some light on this?
