Cap pairings and spectral sequences

Let I be a small category. For an I-diagram XX and Iop-diagrams AA and BB of pointed spaces, each pairing X∧A→BX∧A→B satisfying the projection formula induces a pairing holimiXi∧hocolimiAi→hocolimiBi. In this note we show that there is an induced pairing Erp,q(X)⊗Es,tr(A)→Es−p,q+tr(B) of homotopy spectral sequences compatible with abutments in the sense that FpπqholimiXi∧FsπthocolimiAi→Fs−pπq+thocolimiB.