I read in McCleary's "User's Guide to Spectral Sequences" that a morphism between filtered chain complexes (say $f:X\to Y$, and say that both are as nice as needed, even first quadrant would do for me) induces a morphism of the corresponding spectral sequences, which further on induces a morphism at the infinity page.
However, I can't find in that book (or in the other places I looked) that the corresponding morphism of spectral sequences is compatible with $Hf: HX\to HY$ (in the sense of Weibel, that is, that the induced morphism on the infinity page is same as the one induced by $Hf:HX\to HY$ in the subquotients $F_p H_nX/F_{p-1} H_n X\to F_p H_nY/F_{p-1} H_n Y$). Does this hold, and is this proven somewhere?