Choose a \'etale hypercovering $b : V \to Y$ such that each $V_n = \coprod_{i \in I_n} Y_{n, i}$ with $Y_{n, i}$ affine. This is possible by Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-object} and Remark \ref{hypercovering-remark-take-unions-hypercovering-X} (to replace the hypercovering produced in the lemma by the one having disjoint unions in each degree). Denote $X_{n, i} = Y_{n, i} \times_Y X$ and $U_n = V_n \times_Y X$ so that we obtain an \'etale hypercovering $a : U \to X$ (Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-morphism-sites}) with $U_n = \coprod X_{n, i}$. For each $n, i$ there exists a relative dualizing complex $(\omega_{n, i}^\bullet, \tau_{n, i})$ on $X_{n, i}/Y_{n, i}$. See discussion following Definition \ref{definition-relative-dualizing-proper-flat}. For $\varphi : [m] \to [n]$ and $i \in I_n$ consider the morphisms $g_{\varphi, i} : Y_{n, i} \to Y_{m, \alpha(\varphi)}$ and $g'_{\varphi, i} : X_{n, i} \to X_{m, \alpha(\varphi)}$ which are part of the structure of the given hypercoverings (Hypercoverings, Section \ref{hypercovering-section-hypercovering-sites}). Then we have a unique isomorphisms $$ \iota_{n, i, \varphi} : (L(g'_{n, i})^*\omega_{n, i}^\bullet, Lg_{n, i}^*\tau_{n, i}) \longrightarrow (\omega_{m, \alpha(\varphi)(i)}^\bullet, \tau_{m, \alpha(\varphi)(i)}) $$ of pairs, see discussion following Definition \ref{definition-relative-dualizing-proper-flat}. Observe that $\omega_{n, i}^\bullet$ has vanishing negative self exts on $X_{n, i}$ by Lemma \ref{lemma-relative-dualizing-RHom}. Denote $(\omega_n^\bullet, \tau_n)$ the pair on $U_n/V_n$ constructed using the pairs $(\omega_{n, i}^\bullet, \tau_{n, i})$ for $i \in I_n$. For $\varphi : [m] \to [n]$ and $i \in I_n$ consider the morphisms $g_\varphi : V_n \to V_m$ and $g'_\varphi : U_n \to U_m$ which are part of the structure of the simplicial algebraic spaces $V$ and $U$. Then we have unique isomorphisms $$ \iota_\varphi : (L(g'_\varphi)^*\omega_n^\bullet, Lg_\varphi^*\tau_n) \longrightarrow (\omega_m^\bullet, \tau_m) $$ of pairs constructed from the isomorphisms on the pieces. The uniqueness guarantees that these isomorphisms satisfy the transitivity condition as formulated in Simplicial Spaces, Definition \ref{spaces-simplicial-definition-cartesian-derived-modules}. The assumptions of Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero} are satisfied for $a : U \to X$, the complexes $\omega_n^\bullet$ and the isomorphisms $\iota_\varphi$\footnote{This lemma uses only $\omega_0^\bullet$ and the two maps $\delta_1^1, \delta_0^1 : [1] \to [0]$. The reader can skip the first few lines of the proof of the referenced lemma because here we actually are already given a simplicial system of the derived category of modules.}. Thus we obtain an object $\omega^\bullet$ of $D_\QCoh(\mathcal{O}_X)$ together with an isomorphism $\iota_0 : \omega^\bullet|_{U_0} \to \omega_0^\bullet$ compatible with the two isomorphisms $\iota_{\delta^1_0}$ and $\iota_{\delta^1_1}$. Finally, we apply Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom} to find a unique morphism $$ \tau : Rf_*\omega^\bullet \longrightarrow \mathcal{O}_Y $$ whose restriction to $V_0$ agrees with $\tau_0$; some details omitted -- compare with the end of the  proof of Lemma \ref{lemma-uniqueness-relative-dualizing} for example to see why we have the required vanishing of negative exts. By Lemma \ref{lemma-covering-enough} the pair $(\omega^\bullet, \tau)$ is a relative dualizing complex and the proof is complete.