In this proof unadorned products and tensor products are over $k$. As maps $$ H^p(X, \mathcal{F}) \otimes H^q(Y, \mathcal{G}) \longrightarrow H^n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) $$ we use functoriality of cohomology to get maps $H^p(X, \mathcal{F}) \to H^p(X \times Y, \text{pr}_1^*\mathcal{F})$ and $H^p(Y, \mathcal{G}) \to H^p(X \times Y, \text{pr}_2^*\mathcal{G})$ and then we use the cup product $$ \cup : H^p(X \times Y, \text{pr}_1^*\mathcal{F}) \otimes H^q(X \times Y, \text{pr}_2^*\mathcal{G}) \longrightarrow H^n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) $$ The result is true when $X$ and $Y$ are affine by the vanishing of higher cohomology groups on affines (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}) and the definitions (of pullbacks of quasi-coherent modules and tensor products of quasi-coherent modules). \medskip\noindent Choose finite affine open coverings $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and $\mathcal{V} : Y = \bigcup_{j \in J} V_j$. This determines an affine open covering $\mathcal{W} : X \times Y = \bigcup_{(i, j) \in I \times J} U_i \times V_j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by Cohomology, Lemma \ref{cohomology-lemma-functoriality-cech} we obtain maps $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F}) \quad\text{and}\quad \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_2^*\mathcal{G}) $$ compatible with pullback maps on cohomology. In Cohomology, Equation (\ref{cohomology-equation-needs-signs}) we have constructed a map of complexes $$ \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F}) \otimes \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_2^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) $$ defining the cup product on cohomology. Combining the above we obtain a map of complexes \begin{equation} \label{equation-kunneth-on-cech} \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \end{equation} We warn the reader that this map is not an isomorphism of complexes. Recall that we may compute the cohomologies of our quasi-coherent sheaves using our coverings (Cohomology of Schemes, Lemmas \ref{coherent-lemma-affine-diagonal} and \ref{coherent-lemma-cech-cohomology-quasi-coherent}). Thus on cohomology (\ref{equation-kunneth-on-cech}) reproduces the map of the lemma. \medskip\noindent Consider a short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of quasi-coherent modules. Since the construction of (\ref{equation-kunneth-on-cech}) is functorial in $\mathcal{F}$ and since the formation of the relevant {\v C}ech complexes is exact in the variable $\mathcal{F}$ (because we are taking sections over affine opens) we find a map between short exact sequence of complexes $$ \xymatrix{ \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \ar[r] \ar[d] & \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}') \otimes \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \ar[r] \ar[d] & \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}'') \otimes \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \ar[d] \\ \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F}' \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{W}, \text{pr}_1^*\mathcal{F}'' \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) } $$ (we have dropped the outer zeros). Looking at long exact cohomology sequences we find that if the result of the lemma holds for $2$-out-of-$3$ of $\mathcal{F}, \mathcal{F}', \mathcal{F}''$, then it holds for the third. \medskip\noindent Observe that $X$ has finite cohomological dimension for quasi-coherent modules, see Cohomology of Schemes, Lemma \ref{coherent-lemma-vanishing-nr-affines}. Using induction on $d(\mathcal{F}) = \max \{d \mid H^d(X, \mathcal{F}) \not = 0\}$ we will reduce to the case $d(\mathcal{F}) = 0$. Assume $d(\mathcal{F}) > 0$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-diagonal-universal-delta-functor} we have seen that there exists an embedding $\mathcal{F} \to \mathcal{F}'$ such that $H^p(X, \mathcal{F}') = 0$ for all $p \geq 1$. Setting $\mathcal{F}'' = \Coker(\mathcal{F} \to \mathcal{F}')$ we see that $d(\mathcal{F}'') < d(\mathcal{F})$. Then we can apply the result from the previous paragraph to see that it suffices to prove the lemma for $\mathcal{F}'$ and $\mathcal{F}''$ thereby proving the induction step. \medskip\noindent Arguing in the same fashion for $\mathcal{G}$ we find that we may assume that both $\mathcal{F}$ and $\mathcal{G}$ have nonzero cohomology only in degree $0$. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_V : X \times V = \bigcup_{i \in I} U_i \times V$. It is immediate that $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes \mathcal{G}(V) = \check{\mathcal{C}}^\bullet(\mathcal{U}_V, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) $$ (equality of complexes). We conclude that $$ R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \cong \Gamma(X, \mathcal{F}) \otimes_k \mathcal{G} \cong \bigoplus\nolimits_{\alpha \in A} \mathcal{G} $$ on $Y$. Here $A$ is a basis for the $k$-vector space $\Gamma(X, \mathcal{F})$. Cohomology on $Y$ commutes with direct sums (Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit}). Using the Leray spectral sequence for $\text{pr}_2$ (via Cohomology, Lemma \ref{cohomology-lemma-apply-Leray}) we conclude that $H^n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes H^0(Y, \mathcal{G})$ for $n = 0$. This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree $0$; we omit the verification).