Set $\mathcal{X} = \Complexesstack_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\Sch/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas \ref{lemma-complexes-stack} and \ref{lemma-complexes-diagonal}). Hence it suffices to find a scheme $W$ and a surjective and smooth morphism $W \to \mathcal{X}$. \medskip\noindent Let $B'$ be a scheme and let $B' \to B$ be a surjective \'etale morphism. Set $X' = B' \times_B X$ and denote $f' : X' \to B'$ the projection. Then $\mathcal{X}' = \Complexesstack_{X'/B'}$ is equal to the $2$-fibre product of $\mathcal{X}$ with the category fibred in sets associated to $B'$ over the category fibred in sets associated to $B$ (Remark \ref{remark-complexes-base-change}). By the material in Algebraic Stacks, Section \ref{algebraic-section-representable-properties} the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and \'etale. Hence it suffices to prove the result for $\mathcal{X}'$. In other words, we may assume $B$ is a scheme. \medskip\noindent Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}. Thus we may assume $S = B$. \medskip\noindent Assume $S = B$. Choose an affine open covering $S = \bigcup U_i$. Denote $\mathcal{X}_i$ the restriction of $\mathcal{X}$ to $(\Sch/U_i)_{fppf}$. If we can find schemes $W_i$ over $U_i$ and surjective smooth morphisms $W_i \to \mathcal{X}_i$, then we set $W = \coprod W_i$ and we obtain a surjective smooth morphism $W \to \mathcal{X}$. Thus we may assume $S = B$ is affine. \medskip\noindent Assume $S = B$ is affine, say $S = \Spec(\Lambda)$. Write $\Lambda = \colim \Lambda_i$ as a filtered colimit with each $\Lambda_i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a morphism of algebraic spaces $X_i \to \Spec(\Lambda_i)$ which is proper, flat, of finite presentation and whose base change to $\Lambda$ is $X$. See Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation}, \ref{spaces-limits-lemma-descend-flat}, and \ref{spaces-limits-lemma-eventually-proper}. If we show that $\Complexesstack_{X_i/\Spec(\Lambda_i)}$ is an algebraic stack, then it follows by base change (Remark \ref{remark-complexes-base-change} and Algebraic Stacks, Section \ref{algebraic-section-change-base-scheme}) that $\mathcal{X}$ is an algebraic stack. Thus we may assume that $\Lambda$ is a finite type $\mathbf{Z}$-algebra. \medskip\noindent Assume $S = B = \Spec(\Lambda)$ is affine of finite type over $\mathbf{Z}$. In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin's Axioms, Lemma \ref{artin-lemma-diagonal-representable} to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda$ is a G-ring, see More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-G-ring}. Hence all local rings of $S$ are G-rings. Thus (5) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section \ref{artin-section-axioms}. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas \ref{lemma-complexes-stack}, \ref{lemma-complexes-limits}, \ref{lemma-complexes-RS-star}, \ref{lemma-complexes-tangent-space}. Condition (3) follows from Lemma \ref{lemma-complexes-strong-effectiveness}. Condition (1) is Lemma \ref{lemma-complexes-diagonal}. \medskip\noindent It remains to show condition (4) which is openness of versality. To see this we will use Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}. This follows from Lemma \ref{lemma-complexes-strong-effectiveness} and the proof is complete.