This is probably very well known, something obvious to expect, and written somewhere. But, I do not recall any reference.
Let $F\to X\stackrel{p}{\to} B$ be a fibration where $F$ and $B$ are $(f-1)$-connected and $(b-1)$-connected, respectively. Then, we have Serre exact sequence $$H_{b+f-1}F\to H_{b+f-1}X\to H_{b+f-1}B\stackrel{\tau}{\to}H_{b+f-2}F\to\cdots$$ where $\tau$ is the transgression in the Serre spectral sequence for the above fibration. On the other hand, there is a $(b+f-2)$-equivalence $\Sigma F\to C_p$. So, writing the homology exact sequence of the cofibre sequence yields $$\cdots\to H_jX\to H_j B\to H_jC_p\to\cdots$$ which in the above range, $j<b+f-2$, yields the first exact sequence. Now, can weidentify $\tau$ with the zigzag $$H_jB\to H_jC_p\leftarrow H_j\Sigma F\stackrel{\simeq}{\leftarrow}H_{j-1}F.$$ I would be very grateful for any reference.
