Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition \ref{proposition-quasi-compact-affine-diagonal}. Hence we can identify these categories (and similarly for other quasi-compact algebraic spaces with affine diagonal) and then the statement of the lemma is that the canonical map $\Phi(K) \to Rf_*(K)$ is an isomorphism for all $K$ in $D(\QCoh(\mathcal{O}_X))$. Note that if $K_1 \to K_2 \to K_3 \to K_1[1]$ is a distinguished triangle in $D(\QCoh(\mathcal{O}_X))$ and the statement is true for two-out-of-three, then it is true for the third. \medskip\noindent Let $\mathcal{B} \subset \Ob(X_{spaces, \etale})$ be the set of objects which are quasi-compact and have affine diagonal. For $U \in \mathcal{B}$ and any morphism $g : U \to Z$ where $Z$ is a quasi-compact algebraic space over $S$ with affine diagonal, denote $$ \Phi_g : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_Z)) $$ the derived extension of $g_*$. Let $P(U) =$ ``for any $K$ in $D(\QCoh(\mathcal{O}_U))$ and any $g : U \to Z$ as above the map $\Phi_g(K) \to Rg_*K$ is an isomorphism''. By Remark \ref{remark-how-to} conditions (1), (2), and (3)(a) of Lemma \ref{lemma-induction-principle-separated} hold and we are left with proving (3)(b) and (4). \medskip\noindent Checking condition (3)(b). Let $U$ be an affine scheme \'etale over $X$. Let $g : U \to Z$ be as above. Since the diagonal of $Z$ is affine the morphism $g : U \to Z$ is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-permanence}). Hence $P(U)$ holds by Lemma \ref{lemma-affine-pushforward}. \medskip\noindent Checking condition (4). Let $(U \subset W, V \to W)$ be an elementary distinguished square in $X_{spaces, \etale}$ with $U, W, V$ in $\mathcal{B}$ and $V$ affine. Assume that $P$ holds for $U$, $V$, and $U \times_W V$. We have to show that $P$ holds for $W$. Let $g : W \to Z$ be a morphism to a quasi-compact algebraic space with affine diagonal. Let $K$ be an object of $D(\QCoh(\mathcal{O}_W))$. Consider the distinguished triangle $$ K \to Rj_{U, *}K|_U \oplus Rj_{V, *}K|_V \to Rj_{U \times_W V, *}K|_{U \times_W V} \to K[1] $$ in $D(\mathcal{O}_W)$. By the two-out-of-three property mentioned above, it suffices to show that $\Phi_g(Rj_{U, *}K|_U) \to Rg_*(Rj_{U, *}K|_U)$ is an isomorphism and similarly for $V$ and $U \times_W V$. This is discussed in the next paragraph. \medskip\noindent Let $j : U \to W$ be a morphism $X_{spaces, \etale}$ with $U, W$ in $\mathcal{B}$ and $P$ holds for $U$. Let $g : W \to Z$ be a morphism to a quasi-compact algebraic space with affine diagonal. To finish the proof we have to show that $\Phi_g(Rj_*K) \to Rg_*(Rj_*K)$ is an isomorphism for any $K$ in $D(\QCoh(\mathcal{O}_U))$. Let $\mathcal{I}^\bullet$ be a K-injective complex in $\QCoh(\mathcal{O}_U)$ representing $K$. From $P(U)$ applied to $j$ we see that $j_*\mathcal{I}^\bullet$ represents $Rj_*K$. Since $j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)$ has an exact left adjoint $j^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)$ we see that $j_*\mathcal{I}^\bullet$ is a K-injective complex in $\QCoh(\mathcal{O}_W)$, see Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. Hence $\Phi_g(Rj_*K)$ is represented by $g_*j_*\mathcal{I}^\bullet = (g \circ j)_*\mathcal{I}^\bullet$. By $P(U)$ applied to $g \circ j$ we see that this represents $R_{g \circ j, *}(K) = Rg_*(Rj_*K)$. This finishes the proof.