This is my setting:

Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)\to E_t(X)$$

Then the claim is:

It induces a pairing of Atiyah Hirzebruch Spectral Sequences such that the pairing on the second page $E^{s,-p}_2\otimes E_{s+t,q}^2\to E^2_{t,p+q}$ is the classical cap product in singular (co)homology.

I searched all the literature I know about spectral sequences but this fact is never mentioned

It is not clear to me how to induce such pairing of spectral sequences. I fear it is a generalisation of the fact that we have a pairing $E^{n,-s}_r\otimes E_{n,t}^r\to \mathbb{E}_{s,t}(pt.)$ but since cap product works with classes in (co)homology of different degrees, and produce classes in homology and not only coefficients, I find it hard to generalise the proof of this result. The main problem with that proof is that if I work on the first page, I would need to compare elements in (co)homologies of different spaces, situation which doesn't occur for the traditional pairing.

Are there general results about cap product inducing a pairing of Spectral Sequences?

I'm following Kochman's Bordism, Stable homotopy and Adams Spectral Sequences


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