In the book Fourier-Mukai and Nahm transforms in geometry and mathematical physics page 85 corollary 3.5 there is the following claim:
First some notation.
Let $X$ be an abelian variety of dimension $g$ with dual $Y$. We have the poincare bundle giving a FM functor $$S := D^b(X) \to D^b(Y)$$ and using $P^*$ we have $\hat{S}: D^b(Y) \to D^b(X)$.
The composition is $[-g]$.
Then the claim is that if we take some sheaf $E$ on $X$, and consider the sheaf $F$ on $Y$ defined as $H^0(S(E))$ (this is inner cohomology, not pushforward to a point), then $F$ satisfies that $\hat{S}(F)$ is a complex whose cohomologies (Again inner) vanish except at the index $g$.
It is claimed this follows from the spectral sequence of the composition of $S,\hat{S}$ composing to $[-g]$, but I don't understand how. We have the spectral sequence $\hat{S}^i(S^j(E))$ which converges to $E$ when $i+j=g$ and $0$ otherwise. But there are still many arrows inside touching the column $S^0(E)$ so how does the corollary follow?
Attached is a picture
