I have a concern regarding maps of serre fibrations and the induced map into the respective spectral sequence. The situation is the following.
Consider two serre fibrations $F \rightarrow E \rightarrow B$ and $F' \rightarrow E' \rightarrow B'$. Where $B$ and $B'$ are simply connected to make things easier.
Suppose that we have maps $f: E' \rightarrow E$, and $g: B' \rightarrow B$ that commute with the respective projections. (I couldn't figure out how to make a commutative diagram here) and let $h: F' \rightarrow F$ the map induced in the fibers.
If we consider cohomology with coefficients in a field for instance, the $E_2$ page decomposes as $$E_2 \cong H^*(B) \otimes H^*(F)$$
My first question is, do we have an induced map between the spectral sequences given by
$$g^* \otimes h^* : E_2 \rightarrow E'_2$$
if this is true, is also true that the map induced on the infinite term $E_\infty \rightarrow E'_\infty$ coincides with $f^*: H^*(E)\rightarrow H^*(E')$.
I appreciate any comment.