hcju/stacksqa huggingface dataset:
Being affine is a property of morphisms of schemes which is local on the base and preserved under any base change, see Morphisms, Lemmas \ref{morphisms-lemma-characterize-affine} and \ref{morphisms-lemma-base-change-affine}. Hence Lemma \ref{lemma-descending-types-morphisms} applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{X \to S\}$ flat surjective morphism of affines. Say $X = \Spec(A)$ and $S = \Spec(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is affine. Then $V \to X$ being affine implies that $V = \Spec(B)$ for some $A$-algebra $B$ (see Morphisms, Definition \ref{morphisms-definition-affine}). The isomorphism $\varphi$ corresponds to an isomorphism of rings $$ \varphi^\sharp : B \otimes_R A \longleftarrow A \otimes_R B $$ as $A \otimes_R A$-algebras. The cocycle condition on $\varphi$ says that $$ \xymatrix{ B \otimes_R A \otimes_R A & & A \otimes_R A \otimes_R B \ar[ll] \ar[ld]\\ & A \otimes_R B \otimes_R A \ar[lu] & } $$ is commutative. Inverting these arrows we see that we have a descent datum for modules with respect to $R \to A$ as in Definition \ref{definition-descent-datum-modules}. Hence we may apply Proposition \ref{proposition-descent-module} to obtain an $R$-module $C = \Ker(B \to A \otimes_R B)$ and an isomorphism $A \otimes_R C \cong B$ respecting descent data. Given any pair $c, c' \in C$ the product $cc'$ in $B$ lies in $C$ since the map $\varphi$ is an algebra homomorphism. Hence $C$ is an $R$-algebra whose base change to $A$ is isomorphic to $B$ compatibly with descent data. Applying $\Spec$ we obtain a scheme $U$ over $S$ such that $(V, \varphi) \cong (X \times_S U, can)$ as desired.
Being quasi-affine is a property of morphisms of schemes which is preserved under any base change, see Morphisms, Lemmas \ref{morphisms-lemma-characterize-quasi-affine} and \ref{morphisms-lemma-base-change-quasi-affine}. Hence Lemma \ref{lemma-descending-types-morphisms} applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{X \to S\}$ flat surjective morphism of affines. Say $X = \Spec(A)$ and $S = \Spec(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $\pi : V \to X$ is quasi-affine. \medskip\noindent According to Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine} this means that $$ V \longrightarrow \underline{\Spec}_X(\pi_*\mathcal{O}_V) = W $$ is a quasi-compact open immersion of schemes over $X$. The projections $\text{pr}_i : X \times_S X \to X$ are flat and hence we have $$ \text{pr}_0^*\pi_*\mathcal{O}_V = (\pi \times \text{id}_X)_*\mathcal{O}_{V \times_S X}, \quad \text{pr}_1^*\pi_*\mathcal{O}_V = (\text{id}_X \times \pi)_*\mathcal{O}_{X \times_S V} $$ by flat base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}). Thus the isomorphism $\varphi : V \times_S X \to X \times_S V$ (which is an isomorphism over $X \times_S X$) induces an isomorphism of quasi-coherent sheaves of algebras $$ \varphi^\sharp : \text{pr}_0^*\pi_*\mathcal{O}_V \longrightarrow \text{pr}_1^*\pi_*\mathcal{O}_V $$ on $X \times_S X$. The cocycle condition for $\varphi$ implies the cocycle condition for $\varphi^\sharp$. Another way to say this is that it produces a descent datum $\varphi'$ on the affine scheme $W$ relative to $X$ over $S$, which moreover has the property that the morphism $V \to W$ is a morphism of descent data. Hence by Lemma \ref{lemma-affine} (or by effectivity of descent for quasi-coherent algebras) we obtain a scheme $U' \to S$ with an isomorphism $(W, \varphi') \cong (X \times_S U', can)$ of descent data. We note in passing that $U'$ is affine by Lemma \ref{lemma-descending-property-affine}. \medskip\noindent And now we can think of $V$ as a (quasi-compact) open $V \subset X \times_S U'$ with the property that it is stable under the descent datum $$ can : X \times_S U' \times_S X \to X \times_S X \times_S U', (x_0, u', x_1) \mapsto (x_0, x_1, u'). $$ In other words $(x_0, u') \in V \Rightarrow (x_1, u') \in V$ for any $x_0, x_1, u'$ mapping to the same point of $S$. Because $X \to S$ is surjective we immediately find that $V$ is the inverse image of a subset $U \subset U'$ under the morphism $X \times_S U' \to U'$. Because $X \to S$ is quasi-compact, flat and surjective also $X \times_S U' \to U'$ is quasi-compact flat and surjective. Hence by Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology} this subset $U \subset U'$ is open and we win.
We have seen in Section \ref{section-fpqc-universal-effective-epimorphisms} that representable presheaves are sheaves on the site $(\Sch/S)_\tau$. Moreover, the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) guarantees that maps between representable sheaves correspond one to one with maps between the representing objects. We will use these remarks without further mention during the proof. \medskip\noindent Let us construct the functor from right to left. Let $F$ be a sheaf on $(\Sch/S)_\tau$ such that each $h_{S_i} \times F$ is representable. In this case let $X_i$ be a representing object in $(\Sch/S)_\tau$. It comes equipped with a morphism $X_i \to S_i$. Then both $X_i \times_S S_{i'}$ and $S_i \times_S X_{i'}$ represent the sheaf $h_{S_i} \times F \times h_{S_{i'}}$ and hence we obtain an isomorphism $$ \varphi_{ii'} : X_i \times_S S_{i'} \to S_i \times_S X_{i'} $$ It is straightforward to see that the maps $\varphi_{ii'}$ are morphisms over $S_i \times_S S_{i'}$ and satisfy the cocycle condition. The functor from right to left is given by this construction $F \mapsto (X_i, \varphi_{ii'})$. \medskip\noindent Let us construct a functor from left to right. For each $i$ denote $F_i$ the sheaf $h_{X_i}$. The isomorphisms $\varphi_{ii'}$ give isomorphisms $$ \varphi_{ii'} : F_i \times h_{S_{i'}} \longrightarrow h_{S_i} \times F_{i'} $$ over $h_{S_i} \times h_{S_{i'}}$. Set $F$ equal to the coequalizer in the following diagram $$ \xymatrix{ \coprod_{i, i'} F_i \times h_{S_{i'}} \ar@<1ex>[rr]^-{\text{pr}_0} \ar@<-1ex>[rr]_-{\text{pr}_1 \circ \varphi_{ii'}} & & \coprod_i F_i \ar[r] & F } $$ The cocycle condition guarantees that $h_{S_i} \times F$ is isomorphic to $F_i$ and hence representable. The functor from left to right is given by this construction $(X_i, \varphi_{ii'}) \mapsto F$. \medskip\noindent We omit the verification that these constructions are mutually quasi-inverse functors. The final statements (1) and (2) follow from the constructions.
We only prove (1) and omit the proofs of (2) and (3). As $R \to A$ is faithfully flat, there exists a faithfully flat base change $R \to R'$ such that $R' \to A' = R' \otimes_R A$ has a section (namely take $R' = A$ as in the proof of Lemma \ref{lemma-ff-exact}). Hence, using Lemma \ref{lemma-descent-descends} we may assume that $R \to A$ has a section, say $\sigma : A \to R$. Let $(N, \varphi)$ be a descent datum relative to $R \to A$. Set $$ M = H^0(s(N_\bullet)) = \{n \in N \mid 1 \otimes n = \varphi(n \otimes 1)\} \subset N $$ By Lemma \ref{lemma-recognize-effective} it suffices to show that $A \otimes_R M \to N$ is an isomorphism. \medskip\noindent Take an element $n \in N$. Write $\varphi(n \otimes 1) = \sum a_i \otimes x_i$ for certain $a_i \in A$ and $x_i \in N$. By Lemma \ref{lemma-descent-datum-cosimplicial} we have $n = \sum a_i x_i$ in $N$ (because $\sigma^0_0 \circ \delta^1_1 = \text{id}$ in any cosimplicial object). Next, write $\varphi(x_i \otimes 1) = \sum a_{ij} \otimes y_j$ for certain $a_{ij} \in A$ and $y_j \in N$. The cocycle condition means that $$ \sum a_i \otimes a_{ij} \otimes y_j = \sum a_i \otimes 1 \otimes x_i $$ in $A \otimes_R A \otimes_R N$. We conclude two things from this. First, by applying $\sigma$ to the first $A$ we conclude that $\sum \sigma(a_i) \varphi(x_i \otimes 1) = \sum \sigma(a_i) \otimes x_i$ which means that $\sum \sigma(a_i) x_i \in M$. Next, by applying $\sigma$ to the middle $A$ and multiplying out we conclude that $\sum_i a_i (\sum_j \sigma(a_{ij}) y_j) = \sum a_i x_i = n$. Hence by the first conclusion we see that $A \otimes_R M \to N$ is surjective. Finally, suppose that $m_i \in M$ and $\sum a_i m_i = 0$. Then we see by applying $\varphi$ to $\sum a_im_i \otimes 1$ that $\sum a_i \otimes m_i = 0$. In other words $A \otimes_R M \to N$ is injective and we win.
Part (1) follows from the discussion in Remark \ref{remark-change-topologies-ringed-sites}. Part (2) is just a warning, and can be explained in the following way: First the statement cannot be made precise since $f_*$ does not transform quasi-coherent sheaves into quasi-coherent sheaves in general. Even if this is the case for $f$ (and any base change of $f$), then the compatibility over the big sites would mean that formation of $f_*\mathcal{F}$ commutes with any base change, which does not hold in general. An explicit example is the quasi-compact open immersion $j : X = \mathbf{A}^2_k \setminus \{0\} \to \mathbf{A}^2_k = Y$ where $k$ is a field. We have $j_*\mathcal{O}_X = \mathcal{O}_Y$ but after base change to $\Spec(k)$ by the $0$ map we see that the pushforward is zero. \medskip\noindent Let us prove (3) in case $\tau = \etale$. Note that $f$, and any base change of $f$, transforms quasi-coherent sheaves into quasi-coherent sheaves, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. The equality $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$ means that for any \'etale morphism $g : U \to S$ we have $\Gamma(U, g^*f_*\mathcal{F}) = \Gamma(U \times_S T, (g')^*\mathcal{F})$ where $g' : U \times_S T \to T$ is the projection. This is true by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
We have to show that the category $\mathcal{U}$ described in the proof of Lemma \ref{lemma-exists-hom-0-from-simplicial-set} has a finite subcategory $\mathcal{U}'$ such that the limit of $\mathcal{V}$ over $\mathcal{U}'$ is the same as the limit of $\mathcal{V}$ over $\mathcal{U}$. We will use Categories, Lemma \ref{categories-lemma-initial}. For $m > 0$ let $\mathcal{U}_{\leq m}$ denote the full subcategory with objects $\coprod_{0 \leq n \leq m} U_m$. Let $m_0$ be an integer such that every $n$-simplex of the simplicial set $U$ is degenerate if $n > m_0$. For any $m \geq m_0$ large enough, the subcategory $\mathcal{U}_{\leq m}$ satisfies property (1) of Categories, Definition \ref{categories-definition-initial}. \medskip\noindent Suppose that $u \in U_n$ and $u' \in U_{n'}$ with $n, n' \leq m_0$ and suppose that $\varphi : [k] \to [n]$, $\varphi' : [k] \to [n']$ are morphisms such that $U(\varphi)(u) = U(\varphi')(u')$. A simple combinatorial argument shows that if $k > 2m_0$, then there exists an index $0 \leq i \leq 2m_0$ such that $\varphi(i) =\varphi(i + 1)$ and $\varphi'(i) = \varphi'(i + 1)$. (The pigeon hole principle would tell you this works if $k > m_0^2$ which is good enough for the argument below anyways.) Hence, if $k > 2m_0$, we may write $\varphi = \psi \circ \sigma^{k - 1}_i$ and $\varphi' = \psi' \circ \sigma^{k - 1}_i$ for some $\psi : [k - 1] \to [n]$ and some $\psi' : [k - 1] \to [n']$. Since $s^{k - 1}_i : U_{k - 1} \to U_k$ is injective, see Lemma \ref{lemma-si-injective}, we conclude that $U(\psi)(u) = U(\psi')(u')$ also. Continuing in this fashion we conclude that given morphisms $u \to z$ and $u' \to z$ of $\mathcal{U}$ with $u, u' \in \mathcal{U}_{\leq m_0}$, there exists a commutative diagram $$ \xymatrix{ u \ar[rd] \ar[rrd] & & \\ & a \ar[r] & z \\ u' \ar[ru] \ar[rru] } $$ with $a \in \mathcal{U}_{\leq 2m_0}$. \medskip\noindent It is easy to deduce from this that the finite subcategory $\mathcal{U}_{\leq 2m_0}$ works. Namely, suppose given $x' \in U_n$ and $x'' \in U_{n'}$ with $n, n' \leq 2m_0$ as well as morphisms $x' \to x$ and $x'' \to x$ of $\mathcal{U}$ with the same target. By our choice of $m_0$ we can find objects $u, u'$ of $\mathcal{U}_{\leq m_0}$ and morphisms $u \to x'$, $u' \to x''$. By the above we can find $a \in \mathcal{U}_{\leq 2m_0}$ and morphisms $u \to a$, $u' \to a$ such that $$ \xymatrix{ u \ar[rd] \ar[rrd] \ar[r] & x' \ar[rd] & \\ & a \ar[r] & x \\ u' \ar[ru] \ar[rru] \ar[r] & x'' \ar[ru] & } $$ is commutative. Turning this diagram 90 degrees clockwise we get the desired diagram as in (2) of Categories, Definition \ref{categories-definition-initial}.
By Lemma \ref{lemma-exists-hom-from-simplicial-set-finite} all the required $\Hom$ objects exist and satisfy the correct functorial properties. Now we can identify the $n$th term on the left hand side as the object representing the functor that associates to $X$ the first set of the following sequence of functorial equalities \begin{align*} & \Mor(X \times \Delta[n], \Hom(V, T) \times_{\Hom(U, T)} \Hom(W, T)) \\ & = \Mor(X \times \Delta[n], \Hom(V, T)) \times_{\Mor(X \times \Delta[n], \Hom(U, T))} \Mor(X \times \Delta[n], \Hom(W, T)) \\ & = \Mor(X \times \Delta[n] \times V, T) \times_{\Mor(X \times \Delta[n] \times U, T)} \Mor(X \times \Delta[n] \times W, T) \\ & = \Mor(X \times \Delta[n] \times (V \amalg_U W), T)) \end{align*} Here we have used the fact that $$ (X \times \Delta[n] \times V) \times_{X \times \Delta[n] \times U} (X \times \Delta[n] \times W) = X \times \Delta[n] \times (V \amalg_U W) $$ which is easy to verify term by term. The result of the lemma follows as the last term in the displayed sequence of equalities corresponds to $\Hom(V \amalg_U W, T)_n$.
Under hypothesis (a) we see that the map $f$ preserves the disjoint union decompositions of the splitting of Lemma \ref{lemma-splitting-simplicial-sets}, in other words that we get commutative diagrams $$ \xymatrix{ \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(U_m) \ar[r] \ar[d] & U_n \ar[d] \\ \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(V_m) \ar[r] & V_n. } $$ And then (b) clearly shows that the left vertical arrow is injective (resp.\ surjective, resp.\ bijective).
For any object $A$ of $\mathcal{A}$ we obtain a simplicial abelian group $\Mor_\mathcal{A}(A, U)$. Each of these are canonically split by Lemma \ref{lemma-splitting-simplicial-groups}. Moreover, $$ N(\Mor_\mathcal{A}(A, U_m)) = \bigcap\nolimits_{i = 0}^{m - 1} \Ker(d^m_i) = \Mor_\mathcal{A}(A, N(U_m)). $$ Hence we see that the morphism (\ref{equation-splitting}) becomes an isomorphism after applying the functor $\Mor_\mathcal{A}(A, -)$ for any object of $\mathcal{A}$. Hence it is an isomorphism by the Yoneda lemma.
Pick $m$, $i \leq n$ and some $\varphi : [m] \to [i]$. The image under $U(\psi)$ of $\Im(U(\varphi))$ for any $\psi : [m'] \to [m]$ is equal to the image of $U(\varphi \circ \psi)$ and $\varphi \circ \psi : [m'] \to [i]$. Hence $U'$ is a simplicial object. Pick $m > n$. We have to show $N(U'_m) = 0$. By definition of $N(U_m)$ and $N(U'_m)$ we have $N(U'_m) = U'_m \cap N(U_m)$ (intersection of subobjects). Since $U$ is split by Lemma \ref{lemma-splitting-abelian-category}, it suffices to show that $U'_m$ is contained in the sum $$ \sum\nolimits_{\varphi : [m] \to [m']\text{ surjective}, \ m' < m} \Im(U(\varphi)|_{N(U_{m'})}). $$ By the splitting each $U_{m'}$ is the sum of images of $N(U_{m''})$ via $U(\psi)$ for surjective maps $\psi : [m'] \to [m'']$. Hence the displayed sum above is the same as $$ \sum\nolimits_{\varphi : [m] \to [m']\text{ surjective}, \ m' < m} \Im(U(\varphi)). $$ Clearly $U'_m$ is contained in this by the simple fact that any $\varphi : [m] \to [i]$, $i \leq n$ occurring in the definition of $U'_m$ may be factored as $[m] \to [m'] \to [i]$ with $[m] \to [m']$ surjective and $m' < m$ as in the last displayed sum above.
By Lemma \ref{lemma-trivial-cosk} there are identifications $$ U_i = \lim_{(\Delta/[i])_{\leq n}^{opp}} U(i) $$ for $0 \leq i \leq n$. By Lemma \ref{lemma-formula-limit} we have $$ U_{n + 1} = \lim_{(\Delta/[n + 1])_{\leq n}^{opp}} U(n). $$ Thus we may define for any $\varphi : [i] \to [j]$ with $i, j \leq n + 1$ the corresponding map $\tilde U(\varphi) : \tilde U_j \to \tilde U_i$ exactly as in Lemma \ref{lemma-existence-cosk}. This defines an $(n + 1)$-truncated simplicial object $\tilde U$ with $\text{sk}_n \tilde U = U$. \medskip\noindent To see the adjointness we argue as follows. Given any element $\gamma : \text{sk}_n V \to U$ of the right hand side of the formula consider the morphisms $f_i = \gamma_n \circ d^{n + 1}_i : V_{n + 1} \to V_n \to U_n$. These clearly satisfy the relations $d^n_{j - 1} \circ f_i = d^n_i \circ f_j$ and hence define a unique morphism $V_{n + 1} \to U_{n + 1}$ by our choice of $U_{n + 1}$. Conversely, given a morphism $\gamma' : V \to \tilde U$ of the left hand side we can simply restrict to $\Delta_{\leq n}$ to get an element of the right hand side. We leave it to the reader to show these are mutually inverse constructions.
The statement means that if $U$ is a simplicial object of $\mathcal{C}/X$ which we can think of as a simplicial object of $\mathcal{C}$ with a morphism towards the constant simplicial object $X$, then $\text{cosk}_k U$ computed in $\mathcal{C}/X$ is the same as computed in $\mathcal{C}$. This follows for example from Categories, Lemma \ref{categories-lemma-connected-limit-over-X} because the categories $(\Delta/[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ used in Lemma \ref{lemma-existence-cosk} are connected. Observe that we do not need the categories for $n \leq k$ by Lemma \ref{lemma-trivial-cosk} or Lemma \ref{lemma-recover-cosk}.
This can be seen from the construction of $i_{m!}U$ in Lemma \ref{lemma-left-adjoint-exists}, but we can also argue directly as follows. Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subset with $V'_i = V_i$ for $i \leq m$ and all $i$ simplices degenerate for $i > m$, see Lemma \ref{lemma-simplicial-set-n-skel-sub}. By the adjunction formula, since $\text{sk}_m V' = U$, there is an inverse to the injection $V' \to V$. Hence $V' = V$.
By Lemma \ref{lemma-imshriek-sets} the only nondegenerate simplices of $i_{n!} \text{sk}_n U$ are in degrees $\leq n$. The map $i_{n!} \text{sk}_n U \to U$ is an isomorphism in degrees $\leq n$. Combined we conclude that the map $i_{n!} \text{sk}_n U \to U$ maps nondegenerate simplices to nondegenerate simplices and no two nondegenerate simplices have the same image. Hence Lemma \ref{lemma-injective-map-simplicial-sets} applies. Thus $i_{n!} \text{sk}_n U \to U$ is injective. The result follows easily from this.
Let us denote $\partial \Delta[n] = i_{(n - 1)!} \text{sk}_{n - 1} \Delta[n]$ for convenience. There is a natural map $U \amalg_{\partial \Delta[n]} \Delta[n] \to V$. We have to show that it is bijective in degree $j$ for all $j$. This is clear for $j \leq n$. Let $j > n$. The third condition means that any $z \in V_j$, $z \not \in U_j$ is a degenerate simplex, say $z = s^{j - 1}_i(z')$. Of course $z' \not \in U_{j - 1}$. By induction it follows that $z'$ is a degeneracy of $x$. Thus we conclude that all $j$-simplices of $V$ are either in $U$ or degeneracies of $x$. This implies that the map $U \amalg_{\partial \Delta[n]} \Delta[n] \to V$ is surjective. Note that a nondegenerate simplex of $U \amalg_{\partial \Delta[n]} \Delta[n]$ is either the image of a nondegenerate simplex of $U$, or the image of the (unique) nondegenerate $n$-simplex of $\Delta[n]$. Since clearly $x$ is nondegenerate we deduce that $U \amalg_{\partial \Delta[n]} \Delta[n] \to V$ maps nondegenerate simplices to nondegenerate simplices and is injective on nondegenerate simplices. Hence it is injective, by Lemma \ref{lemma-injective-map-simplicial-sets}.
Let $n$ be the smallest integer such that $V$ has a nondegenerate simplex that does not belong to $U$. Let $x \in V_n$, $x\not \in U_n$ be such a nondegenerate simplex. Let $W \subset V$ be the set of elements which are either in $U$, or are a (repeated) degeneracy of $x$ (in other words, are of the form $V(\varphi)(x)$ with $\varphi : [m] \to [n]$ surjective). It is easy to see that $W$ is a simplicial set. The inclusion $U \subset W$ satisfies the conditions of Lemma \ref{lemma-glue-simplex}. Moreover the number of nondegenerate simplices of $V$ which are not contained in $W$ is exactly one less than the number of nondegenerate simplices of $V$ which are not contained in $U$. Hence we win by induction on this number.
Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subobject of $V$ with $V'_i = V_i$ for $i \leq m$ and $N(V'_i) = 0$ for $i > m$, see Lemma \ref{lemma-simplicial-abelian-n-skel-sub}. By the adjunction formula, since $\text{sk}_m V' = U$, there is an inverse to the injection $V' \to V$. Hence $V' = V$.
By Lemma \ref{lemma-imshriek-abelian} we have $N(i_{n!} \text{sk}_n U)_i = 0$ for $i > n$. The map $i_{n!} \text{sk}_n U \to U$ is an isomorphism in degrees $\leq n$, see Lemma \ref{lemma-recovering-U-for-real}. Combined we conclude that the map $i_{n!} \text{sk}_n U \to U$ induces injective maps $N(i_{n!} \text{sk}_n U)_i \to N(U)_i$ for all $i$. Hence Lemma \ref{lemma-injective-map-simplicial-abelian} applies. Thus $i_{n!} \text{sk}_n U \to U$ is injective. The result follows easily from this.
By Lemma \ref{lemma-morphism-from-coproduct} the object on the left represents the functor which assigns to $X$ the first set of the following equalities \begin{eqnarray*} \Mor(X \times \Delta[n + 1], \text{cosk}_n \text{sk}_n V) & = & \Mor(X \times \text{sk}_n \Delta[n + 1], \text{sk}_n V) \\ & = & \Mor(X \times i_{n !} \text{sk}_n \Delta[n + 1], V). \end{eqnarray*} The object on the right in the formula of the lemma is represented by the functor which assigns to $X$ the last set in the sequence of equalities. This proves the result. \medskip\noindent In the sequence of equalities we have used that $\text{sk}_n (X \times \Delta[n + 1]) = X \times \text{sk}_n \Delta[n + 1]$ and that $i_{n!}(X \times \text{sk}_n \Delta[n + 1]) = X \times i_{n !} \text{sk}_n \Delta[n + 1]$. The first equality is obvious. For any (possibly truncated) simplicial object $W$ of $\mathcal{C}$ and any object $X$ of $\mathcal{C}$ denote temporarily $\Mor_\mathcal{C}(X, W)$ the (possibly truncated) simplicial set $[n] \mapsto \Mor_\mathcal{C}(X, W_n)$. From the definitions it follows that $\Mor(U \times X, W) = \Mor(U, \Mor_\mathcal{C}(X, W))$ for any (possibly truncated) simplicial set $U$. Hence \begin{eqnarray*} \Mor(X \times i_{n !} \text{sk}_n \Delta[n + 1], W) & = & \Mor(i_{n !} \text{sk}_n \Delta[n + 1], \Mor_\mathcal{C}(X, W)) \\ & = & \Mor(\text{sk}_n \Delta[n + 1], \text{sk}_n\Mor_\mathcal{C}(X, W)) \\ & = & \Mor(X \times \text{sk}_n \Delta[n + 1], \text{sk}_nW) \\ & = & \Mor(i_{n!}(X \times \text{sk}_n \Delta[n + 1]), W). \end{eqnarray*} This proves the second equality used, and ends the proof of the lemma.