Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of the projection $R\pi_{Y*}$ should be regarded as "integration along the fibers of Y"?

As a first approximation I know there is the following isomorphism: let $U \subseteq Y$ open

$$\Gamma(U,\pi_{Y*}(\mathcal{G} )) \simeq \bigoplus_{x \in X}\Gamma(U \times \{ x \}, \mathcal{G}_{x})$$

Where $ \mathcal{G}_{x}$ denotes

$$ \mathcal{G}|_{ x \times Y}$$

seen as a sheaf on $Y$.

This actually holds for any sheaf of modules $\mathcal{G}$ over $ X\times Y$, where $X,Y$ are topological spaces and $X$ is equipped with the discrete topology.

Is there a better justification for this interpretation?

Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of the projection $R\pi_{Y*}$ should be regarded as "integration along the fibers"?

As a first approximation I know there is the following isomorphism: let $U \subseteq Y$ open

$$\Gamma(U,\pi_{Y*}(\mathcal{G} )) \simeq \bigoplus_{x \in X}\Gamma(U \times \{ x \}, \mathcal{G}_{x})$$

Where $ \mathcal{G}_{x}$ denotes

$$ \mathcal{G}|_{ x \times Y}$$

seen as a sheaf on $Y$.

This actually holds for any sheaf of modules $\mathcal{G}$ over $ X\times Y$, where $X,Y$ are topological spaces and $X$ is equipped with the discrete topology.

Is there a better justification for this interpretation?