Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. In this proof our simplicial objects and skeleton and coskeleton functors will be taken in this category. \medskip\noindent Choose a polynomial algebra $P_0$ of finite type over $A$ and a surjection $P_0 \to B$. As a first approximation we take $P_\bullet = \text{cosk}_0(P_0)$. In other words, $P_\bullet$ is the simplicial $A$-algebra with terms $P_n = P_0 \times_A \ldots \times_A P_0$. (In the final paragraph of the proof this simplicial object will be denoted $P^0_\bullet$.) By Simplicial, Lemma \ref{simplicial-lemma-cosk-minus-one-equivalence} the map $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets. Also, observe that $P_\bullet = \text{cosk}_0 \text{sk}_0 P_\bullet$. \medskip\noindent Suppose for some $n \geq 0$ we have constructed $P_\bullet$ (in the final paragraph of the proof this will be $P^n_\bullet$) such that \begin{enumerate} \item[(a)] $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets, \item[(b)] $P_k$ is a finitely generated polynomial algebra for $0 \leq k \leq n$, and \item[(c)] $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$ \end{enumerate} By Lemma \ref{lemma-polynomial} we can find a finitely generated polynomial algebra $Q$ over $A$ and a surjection $Q \to P_{n + 1}$. Since $P_n$ is a polynomial algebra the $A$-algebra maps $s_i : P_n \to P_{n + 1}$ lift to maps $s'_i : P_n \to Q$. Set $d'_j : Q \to P_n$ equal to the composition of $Q \to P_{n + 1}$ and $d_j : P_{n + 1} \to P_n$. We obtain a truncated simplicial object $P'_\bullet$ of $\mathcal{A}$ by setting $P'_k = P_k$ for $k \leq n$ and $P'_{n + 1} = Q$ and morphisms $d'_i = d_i$ and $s'_i = s_i$ in degrees $k \leq n - 1$ and using the morphisms $d'_j$ and $s'_i$ in degree $n$. Extend this to a full simplicial object $P'_\bullet$ of $\mathcal{A}$ using $\text{cosk}_{n + 1}$. By functoriality of the coskeleton functors there is a morphism $P'_\bullet \to P_\bullet$ of simplicial objects extending the given morphism of $(n + 1)$-truncated simplicial objects. (This morphism will be denoted $P^{n + 1}_\bullet \to P^n_\bullet$ in the final paragraph of the proof.) \medskip\noindent Note that conditions (b) and (c) are satisfied for $P'_\bullet$ with $n$ replaced by $n + 1$. We claim the map $P'_\bullet \to P_\bullet$ satisfies assumptions (1), (2), (3), and (4) of Simplicial, Lemmas \ref{simplicial-lemma-section} with $n + 1$ instead of $n$. Conditions (1) and (2) hold by construction. By Simplicial, Lemma \ref{simplicial-lemma-cosk-above-object} we see that we have $P_\bullet = \text{cosk}_{n + 1}\text{sk}_{n + 1}P_\bullet$ and $P'_\bullet = \text{cosk}_{n + 1}\text{sk}_{n + 1}P'_\bullet$ not only in $\mathcal{A}$ but also in the category of $A$-algebras, whence in the category of sets (as the forgetful functor from $A$-algebras to sets commutes with all limits). This proves (3) and (4). Thus the lemma applies and $P'_\bullet \to P_\bullet$ is a trivial Kan fibration. By Simplicial, Lemma \ref{simplicial-lemma-trivial-kan-composition} we conclude that $P'_\bullet \to B$ is a trivial Kan fibration and (a) holds as well. \medskip\noindent To finish the proof we take the inverse limit $P_\bullet = \lim P^n_\bullet$ of the sequence of simplicial algebras $$ \ldots \to P^2_\bullet \to P^1_\bullet \to P^0_\bullet $$ constructed above. The map $P_\bullet \to B$ is a trivial Kan fibration by Simplicial, Lemma \ref{simplicial-lemma-limit-trivial-kan}. However, the construction above stabilizes in each degree to a fixed finitely generated polynomial algebra as desired.