Let us denote with $E(X)$ the A-H spectral sequence associated to a CW complex $X$ and homology theory $h_*$: $$ E(X)_{pq}^2 = H_p(X, h_q(\ast))\Longrightarrow h_{p+q}(X)$$ and with $E(Y)$ the one associated to the space $Y$ and same homology theory: $$ E(Y)_{pq}^2 = H_p(Y, h_q(\ast))\Longrightarrow h_{p+q}(Y)$$ Given a map between CW-complex $f \colon X \to Y$ it is known that we have an induced map between A-H spectral sequences $$ f^{sseq} \colon E(X) \to E(Y)$$
I'm interested in applying the Comparison Theorem (Weibel Thm 5.2.12 page 126) (i.e. verify that it can be applied)
My lecturer applied it without checking the hypothesis, since (if I've understood correctly what he said), being $f^{sseq}$ a map of spectral sequences induced by a geometrical map, everything should be fine. I tried working out the details but I have troubles in proving that the map $f^{sseq}$ is compatible with the map $$h_*(f)\colon h_*(X)\to h_*(Y)$$ since according to the definition of compatibility one can find in Weibel, this involves do some checks with the isomorphisms $$ \beta_{pq} \colon E_{pq}^{\infty} \to \dfrac{F_ph_{p+q}}{F_{p-1}h_{p+q}}$$ The problem is that I don't have any idea about how they behave.
Can someone provides me some insights on this application of the Comparison Criterion?