In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant i.e., $x\leq y$ meaning there is an arrow from $y\to x$ and the functor induces morphism $\mathcal{F}(x)\to \mathcal{F}(y)$. For $y\in Y$, define the fibers $$f/y=\{x\in X~|~f(x)\leq y\}.$$ Then there is a Grothendieck spectral sequence $H_i(Y, y\mapsto H_j(f/y,\mathcal{F})) \Longrightarrow H_{i+j}(X,\mathcal{F})$.
I was expecting a similar spectral sequence for cohomology. In order to obtain such, I take $f:X\to Y$ as above and $\mathcal{G}:X\to Ab$ a covariant functor then I can define an induced functor which is also covariant on $Y$ and it maps $y\mapsto H^k(f/y,\mathcal{G})$. My expectation is to get $$H^i(Y,y\mapsto H^j(f/y,\mathcal{G}))\Longrightarrow H^{i+j}(X,\mathcal{G}).$$
Now in order to get such a convergent spectral sequnece I must expect the following to be true. Let $$Ab^X\xrightarrow{L_1} Ab^Y\xrightarrow{L_2} Ab,$$ where $Ab^X$ denotes the category of covariant functors from $X\to Ab$, $L_1$ is induced from $f$ and $L_2$ is $\lim_{\leftarrow Y}$ and the composition must be $\lim_{\leftarrow X}$ such that $L_1$ takes injective objects in $Ab^X$ to $L_2$-acyclic objects. However, being new in this field, I am not very confident whether this is going to work.
I could not find any reference so far where such a spectral cohomology sequence for posets is mentioned. My questions are:
Am I going in the right direction? Should I obtain a cohomology spectral sequence of posets with functor coefficient of such kind out of this and if not why? If such a thing is mentioned (i.e., Gorthendieck spectral sequence of cohomology for posets with functor coefficient) some where in the literature can anyone suggest me a reference?
I apologize for asking too many questions at the same time!!
