Note that the category of abelian presheaves has enough injectives, see Injectives, Proposition \ref{injectives-proposition-presheaves-injectives}. Note that $\check{H}^0(\mathcal{U}, -)$ is a left exact functor from the category of abelian presheaves to the category of $\mathbf{Z}$-modules. Hence the derived functor and the right derived functor exist, see Derived Categories, Section \ref{derived-section-right-derived-functor}. \medskip\noindent Let $\mathcal{I}$ be a injective abelian presheaf. In this case the functor $\Hom_{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ is exact on $\textit{PAb}(\mathcal{C})$. By Lemma \ref{lemma-cech-map-into} we have $$ \Hom_{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet}, \mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}). $$ By Lemma \ref{lemma-homology-complex} we have that $\mathbf{Z}_{\mathcal{U}, \bullet}$ is exact in positive degrees. Hence by the exactness of Hom into $\mathcal{I}$ mentioned above we see that $\check{H}^i(\mathcal{U}, \mathcal{I}) = 0$ for all $i > 0$. Thus the $\delta$-functor $(\check{H}^n, \delta)$ (see Lemma \ref{lemma-cech-cohomology-delta-functor-presheaves}) satisfies the assumptions of Homology, Lemma \ref{homology-lemma-efface-implies-universal}, and hence is a universal $\delta$-functor. \medskip\noindent By Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors} also the sequence $R^i\check{H}^0(\mathcal{U}, -)$ forms a universal $\delta$-functor. By the uniqueness of universal $\delta$-functors, see Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor} we conclude that $R^i\check{H}^0(\mathcal{U}, -) = \check{H}^i(\mathcal{U}, -)$. This is enough for most applications and the reader is suggested to skip the rest of the proof. \medskip\noindent Let $\mathcal{F}$ be any abelian presheaf on $\mathcal{C}$. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ in the category $\textit{PAb}(\mathcal{C})$. Consider the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ with terms $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q)$. Next, consider the total complex $\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))$ associated to this double complex, see Homology, Section \ref{homology-section-double-complexes}. There is a map of complexes $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)) $$ coming from the maps $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^0)$ and there is a map of complexes $$ \check{H}^0(\mathcal{U}, \mathcal{I}^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)) $$ coming from the maps $\check{H}^0(\mathcal{U}, \mathcal{I}^q) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^q)$. Both of these maps are quasi-isomorphisms by an application of Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}. Namely, the columns of the double complex are exact in positive degrees because the {\v C}ech complex as a functor is exact (Lemma \ref{lemma-cech-exact-presheaves}) and the rows of the double complex are exact in positive degrees since as we just saw the higher {\v C}ech cohomology groups of the injective presheaves $\mathcal{I}^q$ are zero. Since quasi-isomorphisms become invertible in $D^{+}(\mathbf{Z})$ this gives the last displayed morphism of the lemma. We omit the verification that this morphism is functorial.