I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:

Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^{p,q}(\mathcal{F})=\mathrm{H}^q(Y,R^q\pi_*\mathcal{F})\implies \mathrm{H}^{p+q}(X,\mathcal{F})$ and $(E')_2^{p,q}(\mathcal{G})\implies \mathrm{H}^{p+q}(X',\mathcal{G})$ their associated Leray spectral sequences (in étale cohomology for some étale sheafs $\mathcal{F},\mathcal{G}$). Assume that there are $f:X'\to X$ and $g:Y'\to Y$ such that the obvious square commutes. Then there is a morphism of spectral sequences $E_r^{p,q}(\mathcal{F})\to (E')^{p,q}_r(g^{-1}\mathcal{F})$ induced by the natural pullback maps on cohomology $E_2^{p,q}(\mathcal{F})\to (E')_2^{p,q}(g^{-1}\mathcal{F})$ such that the map on the infinity-pages agrees with the canonical pullback morphism $\mathrm{H}^n(X,\mathcal{F})\to \mathrm{H}^n(X',f^{-1}\mathcal{F})$.

It would be amazing if someone knows a reliable source for this frequently used result. Thanks in advance!