Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.

The Leray spectral sequence (with complex coefficients) associated with $f$ has $E_2$-term $$E_2^{pq} = H^p(Y,\mathcal{J}^q),$$ where $\mathcal{J}^q$ is the sheafification of the presheaf $$U \to H^q(f^{-1}(U), \mathbb{C}), \quad \forall U \subset Y \text{ open},$$ and converges to $H^*(X,\mathbb{C})$.

In a similar manner, one can construct the Leray spectral sequence associated with $f: (X,A) \to (Y,B)$, seen as a continuous function between pairs. My guess is that it should have $E_2$-term $$E_2^{pq} = H^p(Y,B ; \mathcal{J}^q),$$ where we are now considering local cohomology.

My problem is that I'm not sure to see what the sheaf $\mathcal{J}^q$ looks like in this case. A natural guess would be that it associates to any open set $U \subset Y$ the relative cohomology group $H^q(f^{-1}(U), f^{-1}(U \cap B) ; \mathbb{C})$. In such case, we would have $$\mathcal{J}^q(U) = 0$$ for any open set $U \subset X$ such that $U \subset B$. In particular, the stalk above any point of $B$ would be $0$, which seems very strange.

Could someone help ?

Thanks a lot !