I'm currently trying to understand spectral sequences, and I've started reading "You Could Have Invented Spectral Sequences" By Timothy Chow https://www.ams.org/notices/200601/fea-chow.pdf. In the section titled "Analyzing the Discrepancy" the author states that the "boundaries" of $E^1_{d, 2}:=H_d(E^0_{d, 2})$, where $E^0_{d, p} = C_{d, p}/{C_{d, p-1}}$, are given by
$$B_{d, 2} + C_{d, 1}$$
in other words, the image of $\partial^0:E^0_{d+1, 2}\to E^0_{d, 2}$ is $B_{d, 2} + C_{d, 1}$. What I'm struggling to understand is how you identify the image $\partial^0(C_{d+1, 2}/C_{d+1, 1})\subset C_{d, 2}/C_{d, 1}$ with the subspace $B_{d, 2} + C_{d, 1}$. My main problem is that $B_{d, 2} + C_{d, 1}$ isn't a quotient so how can it be a subspace of $C_{d, 2}/C_{d, 1}$? I thought at first that the notation could be referring to a quotient, as in the elements of $B_{d, 2} + C_{d, 1}$ being the cosets $b + C_{d, 1}$ with $b\in B_{d, 2}$, which would make sense to me. But in the preceding paragraphs the author makes the following identification
$$\frac{Z_d}{B_d}\simeq \frac{Z_d + C_{d, 1}}{B_d + C_{d, 1}}\oplus \frac{Z_d\cap C_{d, 1}}{B_d\cap C_{d, 1}}$$
which, if I understand it correctly, only works if you consider $Z_d + C_{d, 1}$ etc to be the addition of subspaces. So unless the author makes a sudden and unexplained alteration in their notation, I can only conclude that $B_{d, 2} + C_{d, 1}$ is also meant as the addition of subspaces in $C_d$.
So my question is why does
$$\partial^0(E^0_{d+1, 2})\simeq B_{d, 2} + C_{d, 1}$$ hold?
I know there is a similar question on here about this same article, but unfortunately it's talking about slightly different technical points than the one I'm stuck with here, so at the risk of a duplicate question I'm asking about it again.
That isomorphism doesn't hold because the image of $\delta^0$ in $E^0_{d, 2}$ is the quotient $(B_{d,2} + C_{d,1})/ C_{d,1}$. I don't think Chow intends you to take "boundaries" (which he puts in quotes) so literally, in particular, you should not try to perform the identification you mention (since it isn't true).
My interpretation is that you should think of "boundaries" as "the thing you quotient by to get the homology." $E^1_{d,2}$ is literally speaking the quotient $(K/C_{d1})/((B_{d2}+C_{d1})/C_{d1})$ for a certain subspace $K$ containing $C_{d1}$. Applying the third isomorphism theorem, $E^1_{d2} \cong K / (B_{d2} + C_{d1})$, so $B_{d2}+C_{d1}$ is "the thing you quotient by."