I am going through Proposition 2.9 in User's guide in spectral sequences (2nd edition) by McCleary. This is a proof on defining spectral sequences using the language of exact couples. Towards the end of the proof it states

We leave it to the reader to check that the differential

\begin{align} d_r:Z_{r-1}/B_{r-1}\longrightarrow B_r/B_{r-1}\subset Z_{r-1}/B_{r-1} \end{align} is induced on our representation by $j\circ k$ with the proper kernel.

Why is this check necessary? What is the 'proper kernel' supposed to be?



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