In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the theorem "There is the Serre spectral sequence" (theorem 5.2).
Now I would like to have one of the details also for pairs of fibre bundles $(L,L') \xrightarrow{i} (E,E') \rightarrow B$.
Or more precisely, does someone know a reference that states the theorem in (at least) this generality? (if it is even correct)
Theorem: Let $(L,L') \xrightarrow{i} (E,E') \rightarrow B$ be a pair of fibre bundles. There is a first quadrant cohomological spectral sequence of dga's $\{E_r,d_r\}$ with $$ E_2^{p,q} \cong H^p(B;\mathcal{H}^q(L,L')) $$ as algebras, that converges to $H^*(E,E')$. It is natural with respect to maps of pairs of fibrations. Assuming that $(L,L')$ is connected, $$ H^q(E,E') \rightarrow E_{\infty}^{0,q} \xrightarrow{e_L} E_{2}^{0,q} = H^0(B;H^q(L,L')) = H^q(L,L') $$ is the map induced by $i:(L,L') \rightarrow (E,E')$.
Note that I have just translated the condition "$(L,L')$ is connected" directly from the non-relative case where we need "$L$ connected". Not sure what the correct notion here is exactly.
