Spectral-sequence algebra is external to this pair

The Stacks Project proves the comparison between fixed-cover Čech cohomology and sheaf cohomology in the form of a spectral sequence. For a ringed space $X$, an open cover $\mathcal{U}$ of $X$ and a sheaf $\mathcal{F}$ of ${\mathcal{O}}_X$-modules — in particular for the abelian sheaves on a topological space that are used here — there is a spectral sequence $(E_r,d_r{)}_{r\ge 0}$, functorial in $\mathcal{F}$, with $E_2^{p,q}={\check{H}}^p(\mathcal{U},H^q(\mathcal{F}))⟹H^{p+q}(X,\mathcal{F}),$ where $H^q(\mathcal{F})$ denotes the presheaf $V\mapsto H^q(V,\mathcal{F})$ (Stacks Project, Cohomology of Sheaves, Lemma 20.11.5, tag 01ES). The acyclicity hypothesis of the Leray comparison enters there as a degeneration statement: if $H^i(U_{i_0}\cap \cdots \cap U_{i_p},\mathcal{F})=0$ for all $i>0$ and all increasing tuples, then $E_2^{p,q}=0$ for $q\mathrm{\ne }0$, the spectral sequence degenerates at the $E_2$ page, and its edge map is the isomorphism ${\check{H}}^p(\mathcal{U},\mathcal{F})\cong H^p(X,\mathcal{F})$ (ibid., Lemma 20.11.6, tag 01ET).

Nothing of this machinery is constructed or invoked on this page, and the acyclic-cover statement proved here as Leray acyclic-cover comparison does not depend on it. The comparison map itself is built here from the Čech–Godement double complex (Canonical map from fixed-cover Čech to sheaf cohomology), and the acyclic-cover statement is then obtained from Acyclic directions of the Čech–Godement double complex: the two augmentations $u$ and $w$ of the double complex are shown to be quasi-isomorphisms — the first from exactness of the rows, the second from the acyclicity of the columns — by filtering the total complex and applying the mapping-cone criterion, the total complex being formed on finite diagonals.

So the page replaces a spectral-sequence argument by an elementary double-complex argument, and it asserts no spectral-sequence theorem of its own: it neither constructs $E_2$-pages, degenerate pages or convergence, nor appeals to a spectral-sequence theorem proved elsewhere. A consumer who needs the general statement of Lemma 20.11.5 — for instance for a cover that is not acyclic, where the comparison need not be an isomorphism — must take it from a homological-algebra development of spectral sequences; it is cited here only to record where the classical packaging of this comparison lives.