We already know that $\check{H}^n(X, -) \to H^p(X, -)$ is an isomorphism of functors for $n = 0, 1$, see Lemma \ref{lemma-cech-always}. The functors $H^n(X, -)$ form a universal $\delta$-functor, see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}. If we show that $\check{H}^n(X, -)$ forms a universal $\delta$-functor and that $\check{H}^n(X, -) \to H^n(X, -)$ is compatible with boundary maps, then the map will automatically be an isomorphism by uniqueness of universal $\delta$-functors, see Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor}. \medskip\noindent Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of abelian sheaves on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. This gives a complex of complexes $$ 0 \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{H}) \to 0 $$ which is in general not exact on the right. The sequence defines the maps $$ \check{H}^n(\mathcal{U}, \mathcal{F}) \to \check{H}^n(\mathcal{U}, \mathcal{G}) \to \check{H}^n(\mathcal{U}, \mathcal{H}) $$ but isn't good enough to define a boundary operator $\delta : \check{H}^n(\mathcal{U}, \mathcal{H}) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F})$. Indeed such a thing will not exist in general. However, given an element $\overline{h} \in \check{H}^n(\mathcal{U}, \mathcal{H})$ which is the cohomology class of a cocycle $h = (h_{i_0 \ldots i_n})$ we can choose open coverings $$ U_{i_0 \ldots i_n} = \bigcup W_{i_0 \ldots i_n, k} $$ such that $h_{i_0 \ldots i_n}|_{W_{i_0 \ldots i_n, k}}$ lifts to a section of $\mathcal{G}$ over $W_{i_0 \ldots i_n, k}$. By Topology, Lemma \ref{topology-lemma-refine-covering} (this is where we use the assumption that $X$ is hausdorff and quasi-compact) we can choose an open covering $\mathcal{V} : X = \bigcup_{j \in J} V_j$ and $\alpha : J \to I$ such that $V_j \subset U_{\alpha(j)}$ (it is a refinement) and such that for all $j_0, \ldots, j_n \in J$ there is a $k$ such that $V_{j_0 \ldots j_n} \subset W_{\alpha(j_0) \ldots \alpha(j_n), k}$. We obtain maps of complexes $$ \xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{H}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{H}) \ar[r] & 0 } $$ In fact, the vertical arrows are the maps of complexes used to define the transition maps between the {\v C}ech cohomology groups. Our choice of refinement shows that we may choose $$ g_{j_0 \ldots j_n} \in \mathcal{G}(V_{j_0 \ldots j_n}),\quad g_{j_0 \ldots j_n} \longmapsto h_{\alpha(j_0) \ldots \alpha(j_n)}|_{V_{j_0 \ldots j_n}} $$ The cochain $g = (g_{j_0 \ldots j_n})$ is not a cocycle in general but we know that its {\v C}ech boundary $\text{d}(g)$ maps to zero in $\check{\mathcal{C}}^{n + 1}(\mathcal{V}, \mathcal{H})$ (by the commutative diagram above and the fact that $h$ is a cocycle). Hence $\text{d}(g)$ is a cocycle in $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})$. This allows us to define $$ \delta(\overline{h}) = \text{class of }\text{d}(g)\text{ in } \check{H}^{n + 1}(\mathcal{V}, \mathcal{F}) $$ Now, given an element $\xi \in \check{H}^n(X, \mathcal{G})$ we choose an open covering $\mathcal{U}$ and an element $\overline{h} \in \check{H}^n(\mathcal{U}, \mathcal{G})$ mapping to $\xi$ in the colimit defining {\v C}ech cohomology. Then we choose $\mathcal{V}$ and $g$ as above and set $\delta(\xi)$ equal to the image of $\delta(\overline{h})$ in $\check{H}^n(X, \mathcal{F})$. At this point a lot of properties have to be checked, all of which are straightforward. For example, we need to check that our construction is independent of the choice of $\mathcal{U}, \overline{h}, \mathcal{V}, \alpha : J \to I, g$. The class of $\text{d}(g)$ is independent of the choice of the lifts $g_{i_0 \ldots i_n}$ because the difference will be a coboundary. Independence of $\alpha$ holds\footnote{This is an important check because the nonuniqueness of $\alpha$ is the only thing preventing us from taking the colimit of {\v C}ech complexes over all open coverings of $X$ to get a short exact sequence of complexes computing {\v C}ech cohomology.} because a different choice of $\alpha$ determines homotopic vertical maps of complexes in the diagram above, see Section \ref{section-refinements-cech}. For the other choices we use that given a finite collection of coverings of $X$ we can always find a covering refining all of them. We also need to check additivity which is shown in the same manner. Finally, we need to check that the maps $\check{H}^n(X, -) \to H^n(X, -)$ are compatible with boundary maps. To do this we choose injective resolutions $$ \xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar[d] & \mathcal{G} \ar[r] \ar[d] & \mathcal{H} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{I}_1^\bullet \ar[r] & \mathcal{I}_2^\bullet \ar[r] & \mathcal{I}_3^\bullet \ar[r] & 0 } $$ as in Derived Categories, Lemma \ref{derived-lemma-injective-resolution-ses}. This will give a commutative diagram $$ \xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_1^\bullet)) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_2^\bullet)) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_3^\bullet)) \ar[r] & 0 } $$ Here $\mathcal{U}$ is an open covering as above and the vertical maps are those used to define the maps $\check{H}^n(\mathcal{U}, -) \to H^n(X, -)$, see Lemma \ref{lemma-cech-cohomology}. The bottom complex is exact as the sequence of complexes of injectives is termwise split exact. Hence the boundary map in cohomology is computed by the usual procedure for this lower exact sequence, see Homology, Lemma \ref{homology-lemma-long-exact-sequence-cochain}. The same will be true after passing to the refinement $\mathcal{V}$ where the boundary map for {\v C}ech cohomology was defined. Hence the boundary maps agree because they use the same construction (whenever the first one is defined on an element in {\v C}ech cohomology on a given covering). This finishes our discussion of the construction of the structure of a $\delta$-functor on {\v C}ech cohomology and why this structure is compatible with the given $\delta$-functor structure on usual cohomology. \medskip\noindent Finally, we may apply Lemma \ref{lemma-injective-trivial-cech} to see that higher {\v C}ech cohomology is trivial on injective sheaves. Hence we see that {\v C}ech cohomology is a universal $\delta$-functor by Homology, Lemma \ref{homology-lemma-efface-implies-universal}.