If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal coefficient theorem for cohomology.

Now, suppose I have a fibration $F \to E \to B$ such that $F$ and $B$ have homology and cohomology over $k$ of finite type in each degree and $\pi_1(B) = 0$ for simplicity. Certainly, the $E_2$-page of the cohomology Serre spectral sequence will be dual to the $E^2$-page homology Serre spectral sequence. My first question is: Is it also true that the differentials for the cohomology spectral sequence are dual as a linear map to the differentials of the homology spectral sequence, and vice versa?

Secondly, the cohomology Serre spectral sequence is a multiplicative one. Is the homology one comultiplicative? If so, is the product for cohomology dual to the coproduct for homology?

Finally, if all of this holds, to which extend can it generalized?