hcju/stacksqa huggingface dataset:
From the construction of Lemma \ref{lemma-presentation-construction} we have a commutative diagram $$ \xymatrix{ R = U \times_{f, \mathcal{F}, f} U \ar[r]_-s \ar[d]_t & U \ar[d]^f \\ U \ar[r]^f & \mathcal{F} } $$ where $t, s$ are the first and second projections.  So $t, s$ are smooth by Lemma \ref{lemma-smooth-properties}.  Hence (1) holds. \medskip\noindent If the assumption of (2) holds, then by Lemma \ref{lemma-smooth-morphism-essentially-surjective} the morphism $f : U \to \mathcal{F}$ is essentially surjective. Hence by Lemma \ref{lemma-presentation-morphism} the morphism $[f] : [U/R] \to \mathcal{F}$ is an equivalence.
Note that $U$ is a deformation functor by Example \ref{example-prorepresentable-deformation-functor}. By Lemma \ref{lemma-deformation-functor-diagonal} we see that $R = U \times_{f, \mathcal{F}, f} U$ is a deformation functor whose tangent space $TR = T(U \times_{f, \mathcal{F}, f} U)$ sits in an exact sequence $0 \to \text{Inf}(\mathcal{F}) \to TR \to TU \oplus TU$. Since we have assumed the first space has finite dimension and since $TU$ has finite dimension by Example \ref{example-tangent-space-prorepresentable-functor} we see that $\dim TR < \infty$. The map $\gamma : \text{Der}_\Lambda(k, k) \to TR$ see (\ref{equation-map}) is injective because its composition with $TR \to TU$ is injective by Theorem \ref{theorem-Schlessinger-prorepresentability} for the prorepresentable functor $U$. Thus $R$ is prorepresentable by Theorem \ref{theorem-Schlessinger-prorepresentability}. It follows from Lemma \ref{lemma-groupoid-in-functors-prorep-equivalences} that $(U, R, s, t, c)$ is prorepresentable.
Part (1) follows immediately from the definitions. To see part (2) set $\mathcal{F} = [U/R]$. Since $\mathcal{F}$ has a presentation it is a deformation category, see Theorem \ref{theorem-presentation-deformation-groupoid}. In particular it satisfies (RS), (S1), and (S2), see Lemma \ref{lemma-RS-implies-S1-S2}. Recall that minimal versal formal objects are unique up to isomorphism, see Lemma \ref{lemma-minimal-versal}. By Theorem \ref{theorem-miniversal-object-existence} a minimal versal object induces a map $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ satisfying (\ref{equation-bijective-orbits}). Since $U \cong \underline{R}|_{\mathcal{C}_\Lambda}$ over $\mathcal{F}$ we see that $TU \to T\mathcal{F} = T[U/R]$ satisfies the property as stated in the lemma.
If $U \cong \underline{R}|_{\mathcal{C}_\Lambda}$ for some $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$, then completing $\varphi$ gives a morphism $\underline{R} \to \underline{R}$. If $f: R \to R$ is the corresponding morphism in $\widehat{\mathcal{C}}_\Lambda$, then $f$ induces an isomorphism $\text{Der}_\Lambda(R, k) \to \text{Der}_\Lambda(R, k)$, see Example \ref{example-tangent-space-map-prorepresentable-functor}. In particular $f$ is a surjection by Lemma \ref{lemma-derivations-surjective}. As a surjective endomorphism of a Noetherian ring is an isomorphism (see Algebra, Lemma \ref{algebra-lemma-surjective-endo-noetherian-ring-is-iso}) we conclude $f$, hence $\underline{R} \to \underline{R}$, hence $\varphi : U \to U$ is an isomorphism.
A morphism $\varphi : [U/R] \to [U/R]$ is the same thing as a morphism $\varphi : (U, R, s, t, c) \to (U, R, s, t, c)$ of groupoids in functors over $\mathcal{C}_\Lambda$ as defined in Definition \ref{definition-groupoid-in-functors}. Denote $\phi : U \to U$ and $\psi : R \to R$ the corresponding morphisms. Because the diagram $$ \xymatrix{ & \text{Der}_\Lambda(k, k) \ar[dr]_\gamma \ar[dl]^\gamma \\ TU \ar[rr]_{d\phi} \ar[d] & & TU \ar[d]  \\ T[U/R] \ar[rr]^{d\varphi} & & T[U/R] } $$ is commutative, since $d\varphi$ is bijective, and since we have the characterization of minimality in Lemma \ref{lemma-characterize-minimal-groupoid-in-functors} we conclude that $d\phi$ is injective (hence bijective by dimension reasons). Thus $\phi : U \to U$ is an isomorphism by Lemma \ref{lemma-surjective-morphism-prorepresentable-functor}. We can use a similar argument, using the exact sequence $$ 0 \to \text{Inf}([U/R]) \to TR \to TU \oplus TU $$ of Lemma \ref{lemma-deformation-functor-diagonal} to prove that $\psi : R \to R$ is an isomorphism. But is also a consequence of the fact that $R = U \times_{[U/R]} U$ and that $\varphi$ and $\phi$ are isomorphisms.
Let $\psi : [U'/R'] \to [U/R]$ be a quasi-inverse to $\varphi$. Then $\psi \circ \varphi$ and $\varphi \circ \psi$ are isomorphisms by Lemma \ref{lemma-minimal-prorepresentable-groupoid-autoequivalence}, hence $\varphi$ and $\psi$ are isomorphisms.
As $\mathcal{F}$ is a deformation category it satisfies (S1) and (S2), see Lemma \ref{lemma-RS-implies-S1-S2}. By Lemma \ref{lemma-versal-object-existence} there exists a versal formal object. By Lemma \ref{lemma-minimal-versal} there exists a minimal versal formal object $\xi/R$ as in the statement of the lemma. Setting $U = \underline{R}|_{\mathcal{C}_\Lambda}$ the associated map $\underline{\xi} : U \to \mathcal{F}$ is smooth (this is the definition of a versal formal object). Let $(U, R, s, t, c)$ be the groupoid in functors constructed in Lemma \ref{lemma-presentation-construction} from the map $\underline{\xi}$. By Lemma \ref{lemma-smooth-groupoid-in-functors-construction} we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that $[U/R] \to \mathcal{F}$ is an equivalence. By Lemma \ref{lemma-prorepresentable-groupoid-in-functors-construction} we see that $(U, R, s, t, c)$ is prorepresentable. Finally, $(U, R, s, t, c)$ is minimal because $U \to [U/R] = \mathcal{F}$ corresponds to the minimal versal formal object $\xi$.
Suppose we have any $r \geq 1$ and two maps $h_1, h_2 : R \to R[[t_1, \ldots, t_r]]$ such that $h_1(a) \bmod (t_1, \ldots, t_r) =  h_2(a) \bmod (t_1, \ldots, t_r) = a$ for all $a \in R$ and a map $k : R[[t_1, \ldots, t_r]] \to S$. Then we claim $k \circ h_1$ is formally homotopic to $k \circ h_2$. The symmetric inherent in this claim will show that our notion of formally homotopic is symmetric. Namely, the map $$ \Psi : R[[t_1, \ldots, t_r]] \longrightarrow R[[t_1, \ldots, t_r]],\quad \sum a_I t^I \longmapsto \sum h_1(a_I)t^I $$ is an isomorphism. Set $h(a) = \Psi^{-1}(h_2(a))$ for $a \in R$ and $k' = k \circ \Psi$, then we see that $(r, h, k')$ is a formal homotopy between $k \circ h_1$ and $k \circ h_2$, proving the claim \medskip\noindent Say we have three maps $f_1, f_2, f_3 : R \to S$ as above and a formal homotopy $(r_1, h_1, k_1)$ between $f_1$ and $f_2$ and a formal homotopy $(r_2, h_2, k_2)$ between $f_3$ and $f_2$ (!). After relabeling the coordinates we may assume $h_2 : R \to R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]]$ and $k_2 : R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \to S$. By choosing a suitable isomorphism $$ R[[t_1, \ldots, t_{r_1 + r_2}]] \longrightarrow R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \widehat{\otimes}_{h_2, R, h_1} R[[t_1, \ldots, t_{r_1}]] $$ we may fit these maps into a commutative diagram $$ \xymatrix{ R \ar[r]_{h_1} \ar[d]^{h_2} & R[[t_1, \ldots, t_{r_1}]] \ar[d]^{h_2'} \\ R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \ar[r]^{h_1'} & R[[t_1, \ldots, t_{r_1 + r_2}]] } $$ with $h_2'(t_i) = t_i$ for $1 \leq i \leq r_1$ and $h_1'(t_i) = t_i$ for $r_1 + 1 \leq i \leq r_2$. Some details omitted. Since this diagram is a pushout in the category $\widehat{\mathcal{C}}_\Lambda$ (see proof of Lemma \ref{lemma-CLambdahat-pushouts}) and since $k_1 \circ h_1 = f_2 = k_2 \circ h_2$ we conclude there exists a map $$ k : R[[t_1, \ldots, t_{r_1 + r_2}]] \to S $$ with $k_1 = k \circ h_2'$ and $k_2 = k \circ h_1'$. Denote $h = h_1' \circ h_2 = h_2' \circ h_1$. Then we have \begin{enumerate} \item $k(h_1'(a)) = k_2(a) = f_3(a)$, and \item $k(h_2'(a)) = k_1(a) = f_1(a)$. \end{enumerate} By the claim in the first paragraph of the proof this shows that $f_1$ and $f_3$ are formally homotopic.
By Theorem \ref{theorem-presentation-deformation-groupoid} and its proof, $\mathcal{F}$ has a presentation by a smooth prorepresentable groupoid $$ (\underline{R}, \underline{R_1}, s, t, c, e, i)|_{\mathcal{C}_\Lambda} $$ in functors on $\mathcal{C}_\lambda$ such that $\mathcal{F}$. Then the maps $s : R \to R_1$ and $t : R \to R_1$ are formally smooth ring maps and $e : R_1 \to R$ is a section. In particular, we can choose an isomorphism $R_1 = R[[t_1, \ldots, t_r]]$ for some $r \geq 0$ such that $s$ is the embedding $R \subset R[[t_1, \ldots, t_r]]$ and $t$ corresponds to a map $h : R \to R[[t_1, \ldots, t_r]]$ with $h(a) \bmod (t_1, \ldots, t_r) = a$ for all $a \in R$. The existence of the isomorphism $\alpha : f_*\xi \to g_*\xi$ means exactly that there is a map $k : R_1 \to S$ such that $f = k \circ s$ and $g = k \circ t$. This exactly means that $(r, h, k)$ is a formal homotopy between $f$ and $g$.
Part (1) is immediate from the definitions. \medskip\noindent Since $\mathcal{F}_{l/k}(l) = \mathcal{F}(k)$ part (2) follows from the definition, see Definition \ref{definition-predeformation-category}. \medskip\noindent Part (3) follows as the functor (\ref{equation-comparison}) commutes with fibre products and transforms surjective maps into surjective maps, see Definition \ref{definition-S1-S2}. \medskip\noindent Part (4). To see this consider a diagram $$ \xymatrix{           & l[\epsilon] \ar[d] \\ B  \ar[r] & l } $$ in $\mathcal{C}_{\Lambda, l}$ as in Definition \ref{definition-S1-S2}. Applying the functor (\ref{equation-comparison}) we obtain $$ \xymatrix{           & k[l\epsilon] \ar[d] \\ B \times_l k  \ar[r] & k } $$ where $l\epsilon$ denotes the finite dimensional $k$-vector space $l\epsilon \subset l[\epsilon]$. According to Lemma \ref{lemma-S2-extensions} the condition of (S2) for $\mathcal{F}$ also holds for this diagram. Hence (S2) holds for $\mathcal{F}_{l/k}$. \medskip\noindent Part (5) follows from the characterization of (RS) in Lemma \ref{lemma-RS-2-categorical} part (2) and the fact that (\ref{equation-comparison}) commutes with fibre products.
By Lemma \ref{lemma-elementary-properties-change-of-field} we may replace $\mathcal{F}$ by $\overline{\mathcal{F}}$. Moreover we see that $T\mathcal{F}$, resp.\ $T\mathcal{F}_{l/k}$ has a canonical $k$-vector space structure, resp.\ $l$-vector space structure, see Lemma \ref{lemma-tangent-space-vector-space}. Then $$ T\mathcal{F}_{l/k} = \mathcal{F}_{l/k}(l[\epsilon]) = \mathcal{F}(k[l\epsilon]) = T\mathcal{F} \otimes_k l $$ the last equality by Lemma \ref{lemma-tangent-space-vector-space}. More generally, given a finite dimensional $l$-vector space $V$ we have $$ \mathcal{F}_{l/k}(l[V]) = \mathcal{F}(k[V_k]) = T\mathcal{F} \otimes_k V_k $$ where $V_k$ denotes $V$ seen as a $k$-vector space. We conclude that the functors $V \mapsto \mathcal{F}_{l/k}(l[V])$ and $V \mapsto T\mathcal{F} \otimes_k V_k$ are canonically identified as functors to the category of sets. By Lemma \ref{lemma-linear-functor} we see there is at most one way to turn either functor into an $l$-linear functor. Hence the isomorphisms are compatible with the $l$-vector space structures and we win.
Let $x_0 \in \Ob(\mathcal{F}(k))$ and denote $x_{l, 0}$ the corresponding object of $\mathcal{F}_{l/k}$ over $l$. Recall that $\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})$ and $\text{Inf}(\mathcal{F}_{l/k}) = \text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$, see Remark \ref{remark-trivial-aut-point}. Recall that the vector space structure on $\text{Inf}_{x_0}(\mathcal{F})$ comes from identifying it with the tangent space of the functor $\mathit{Aut}(x_0)$ which is defined on the category $\mathcal{C}_{k, k}$ of Artinian local $k$-algebras with residue field $k$. Similarly, $\text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$ is the tangent space of $\mathit{Aut}(x_{l, 0})$ which is defined on the category $\mathcal{C}_{l, l}$ of Artinian local $l$-algebras with residue field $l$. Unwinding the definitions we see that $\mathit{Aut}(x_{l, 0})$ is the restriction of $\mathit{Aut}(x_0)_{l/k}$ (which lives on $\mathcal{C}_{k, l}$) to $\mathcal{C}_{l, l}$. Since there is no difference between the tangent space of $\mathit{Aut}(x_0)_{l/k}$ seen as a functor on $\mathcal{C}_{k, l}$ or $\mathcal{C}_{l, l}$, the lemma follows from  Lemma \ref{lemma-tangent-space-change-of-field} and the fact that $\mathit{Aut}(x_0)$ satisfies (RS) by Lemma \ref{lemma-Aut-functor-RS} (whence we have (S2) by Lemma \ref{lemma-RS-implies-S1-S2}).
Construction of $S$. Choose a surjection $R[x_1, \ldots, x_n] \to l$ of $R$-algebras. The kernel is a maximal ideal $\mathfrak m$. Set $S$ equal to the $\mathfrak m$-adic completion of the Noetherian ring $R[x_1, \ldots, x_n]$. Then $S$ is in $\widehat{\mathcal{C}}_{\Lambda, l}$ by Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian}. The map $R \to S$ is formally smooth in the $\mathfrak m_S$-adic topology by More on Algebra, Lemmas \ref{more-algebra-lemma-formally-smooth} and \ref{more-algebra-lemma-formally-smooth-completion} and the fact that $R \to R[x_1, \ldots, x_n]$ is formally smooth. (Compare with the proof Lemma \ref{lemma-exists-smooth}.) \medskip\noindent Since $\xi$ is versal, the transformation $\underline{\xi} : \underline{R}|_{\mathcal{C}_{\Lambda, k}} \to \mathcal{F}$ is smooth. By Lemma \ref{lemma-change-of-fields-smooth} the induced map $$ (\underline{R}|_{\mathcal{C}_{\Lambda, k}})_{l/k} \longrightarrow \mathcal{F}_{l/k} $$ is smooth. Thus it suffices to construct a smooth morphism $\underline{S}|_{\mathcal{C}_{\Lambda, l}} \to (\underline{R}|_{\mathcal{C}_{\Lambda, k}})_{l/k}$. To give such a map means for every object $B$ of $\mathcal{C}_{\Lambda, l}$ a map of sets $$ \Mor_{\widehat{\mathcal{C}}_{\Lambda, l}}(S, B) \longrightarrow \Mor_{\widehat{\mathcal{C}}_{\Lambda, k}}(R, B \times_l k) $$ functorial in $B$. Given an element $\varphi : S \to B$ on the left hand side we send it to the composition $R \to S \to B$ whose image is contained in the sub $\Lambda$-algebra $B \times_l k$. Smoothness of the map means that given a surjection $B' \to B$ and a commutative diagram $$ \xymatrix{ S \ar[r] & B \ar@{=}[r] & B \\ R \ar[u] \ar[r] & B' \times_l k \ar[u] \ar[r] & B' \ar[u] } $$ we have to find a ring map $S \to B'$ fitting into the outer rectangle. The existence of this map is guaranteed as we chose $R \to S$ to be formally smooth in the $\mathfrak m_S$-adic topology, see More on Algebra, Lemma \ref{more-algebra-lemma-lift-continuous}.
Choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$ where $\underline{M} \to \underline{N}$ is given by $\varphi : M \to N$. By Lemma \ref{lemma-adequate-finite-presentation} it suffices to construct $L \to F$ such that $L(B) \to F(B)$ is surjective for every finitely presented $A$-algebra $B$. Hence it suffices to construct, given a finitely presented $A$-algebra $B$ and an element $\xi \in F(B)$ a map $L \to F$ with $L$ linearly adequate such that $\xi$ is in the image of $L(B) \to F(B)$. (Because there is a set worth of such pairs $(B, \xi)$ up to isomorphism.) \medskip\noindent To do this write $\sum_{i = 1, \ldots, n} m_i \otimes b_i$ the image of $\xi$ in $\underline{M}(B) = M \otimes_A B$. We know that $\sum \varphi(m_i) \otimes b_i = 0$ in $N \otimes_A B$. As $N$ is a filtered colimit of finitely presented $A$-modules, we can find a finitely presented $A$-module $N'$, a commutative diagram of $A$-modules $$ \xymatrix{ A^{\oplus n} \ar[r] \ar[d]_{m_1, \ldots, m_n} & N' \ar[d] \\ M \ar[r] & N } $$ such that $(b_1, \ldots, b_n)$ maps to zero in $N' \otimes_A B$. Choose a presentation $A^{\oplus l} \to A^{\oplus k} \to N' \to 0$. Choose a lift $A^{\oplus n} \to A^{\oplus k}$ of the map $A^{\oplus n} \to N'$ of the diagram. Then we see that there exist $(c_1, \ldots, c_l) \in B^{\oplus l}$ such that $(b_1, \ldots, b_n, c_1, \ldots, c_l)$ maps to zero in $B^{\oplus k}$ under the map $B^{\oplus n} \oplus B^{\oplus l} \to B^{\oplus k}$. Consider the commutative diagram $$ \xymatrix{ A^{\oplus n} \oplus A^{\oplus l} \ar[r] \ar[d] & A^{\oplus k} \ar[d] \\ M \ar[r] & N } $$ where the left vertical arrow is zero on the summand $A^{\oplus l}$. Then we see that $L$ equal to the kernel of $\underline{A^{\oplus n + l}} \to \underline{A^{\oplus k}}$ works because the element $(b_1, \ldots, b_n, c_1, \ldots, c_l) \in L(B)$ maps to $\xi$.
Suppose that the map $A^{\oplus n} \to A^{\oplus m}$ is given by the $m \times n$-matrix $(a_{ij})$. Consider the ring $B = A[x_1, \ldots, x_n]/(\sum a_{ij}x_j)$. The element $(x_1, \ldots, x_n) \in \underline{A^{\oplus n}}(B)$ maps to zero in $\underline{A^{\oplus m}}(B)$ hence is the image of a unique element $\xi \in L(B)$. Note that $\xi$ has the following universal property: for any $A$-algebra $C$ and any $\xi' \in L(C)$ there exists an $A$-algebra map $B \to C$ such that $\xi$ maps to $\xi'$ via the map $L(B) \to L(C)$. \medskip\noindent Note that $B$ is a graded $A$-algebra, hence we can use Lemmas \ref{lemma-flat-functor-split} and \ref{lemma-adequate-flat} to decompose the values of our functors on $B$ into graded pieces. Note that $\xi \in L(B)^{(1)}$ as $(x_1, \ldots, x_n)$ is an element of degree one in $\underline{A^{\oplus n}}(B)$. Hence we see that $\varphi(\xi) \in \underline{M}(B)^{(1)} = M \otimes_A B_1$. Since $B_1$ is generated by $x_1, \ldots, x_n$ as an $A$-module we can write $\varphi(\xi) = \sum m_i \otimes x_i$. Consider the map $A^{\oplus n} \to M$ which maps the $i$th basis vector to $m_i$. By construction the associated map $\underline{A^{\oplus n}} \to \underline{M}$ maps the element $\xi$ to $\varphi(\xi)$. It follows from the universal property mentioned above that the diagram commutes.
By Lemma \ref{lemma-adequate-surjection-from-linear} we may assume that $F = \bigoplus L_i$ is a direct sum of linearly adequate functors. Choose exact sequences $0 \to L_i \to \underline{A^{\oplus n_i}} \to \underline{A^{\oplus m_i}}$. For each $i$ choose a map $A^{\oplus n_i} \to M$ as in Lemma \ref{lemma-lift-map}. Consider the diagram $$ \xymatrix{ 0 \ar[r] & \bigoplus L_i  \ar[r] \ar[d] & \bigoplus \underline{A^{\oplus n_i}} \ar[r] \ar[ld] & \bigoplus \underline{A^{\oplus m_i}} \\ & \underline{M} } $$ Consider the $A$-modules $$ Q = \Coker(\bigoplus A^{\oplus n_i} \to M \oplus \bigoplus A^{\oplus m_i}) \quad\text{and}\quad P = \Coker(\bigoplus A^{\oplus n_i} \to \bigoplus A^{\oplus m_i}). $$ Then we see that $\Coker(\varphi)$ is isomorphic to the kernel of $\underline{Q} \to \underline{P}$.
Choose an injection $G \to \underline{M}$. Then we have an injection $G/F \to \underline{M}/F$. By Lemma \ref{lemma-cokernel-into-module} we see that $\underline{M}/F$ is adequate, hence we can find an injection $\underline{M}/F \to \underline{N}$. Composing we obtain an injection $G/F \to \underline{N}$. By Lemma \ref{lemma-cokernel-into-module} the cokernel of the induced map $G \to \underline{N}$ is adequate hence we can find an injection $\underline{N}/G \to \underline{K}$. Then $0 \to G/F \to \underline{N} \to \underline{K}$ is exact and we win.
Choose an injection $F \to \underline{M}$ and an injection $G \to \underline{N}$. Denote $F \to \underline{M \oplus N}$ the diagonal map so that $$ \xymatrix{ F \ar[d] \ar[r] & G \ar[d] \\ \underline{M \oplus N} \ar[r] & \underline{N} } $$ commutes. By Lemma \ref{lemma-cokernel-adequate} we can find a module map $M \oplus N \to K$ such that $F$ is the kernel of $\underline{M \oplus N} \to \underline{K}$. Then $\Ker(\varphi)$ is the kernel of $\underline{M \oplus N} \to \underline{K \oplus N}$.
Choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$. If we can show that $(\underline{M} \oplus G)/F$ is adequate, then $G$ is the kernel of the map of adequate functors $(\underline{M} \oplus G)/F \to \underline{N}$, hence adequate by Lemma \ref{lemma-kernel-adequate}. Thus we may assume $F = \underline{M}$. \medskip\noindent We can choose a surjection $L \to H$ where $L$ is a direct sum of linearly adequate functors, see Lemma \ref{lemma-adequate-surjection-from-linear}. If we can show that the pullback $G \times_H L$ is adequate, then $G$ is the cokernel of the map $\Ker(L \to H) \to G \times_H L$ hence adequate by Lemma \ref{lemma-cokernel-adequate}. Thus we may assume that $H = \bigoplus L_i$ is a direct sum of linearly adequate functors. By Lemma \ref{lemma-extension-adequate-key} each of the pullbacks $G \times_H L_i$ is adequate. By Lemma \ref{lemma-colimit-adequate} we see that $\bigoplus G \times_H L_i$ is adequate. Then $G$ is the cokernel of $$ \bigoplus\nolimits_{i \not = i'} F \longrightarrow \bigoplus G \times_H L_i $$ where $\xi$ in the summand $(i, i')$ maps to $(0, \ldots, 0, \xi, 0, \ldots, 0, -\xi, 0, \ldots, 0)$ with nonzero entries in the summands $i$ and $i'$. Thus $G$ is adequate by Lemma \ref{lemma-cokernel-adequate}.
The functors $B \to F(B \otimes_A A')$ and $B \mapsto F(B \otimes_A A' \otimes_A A')$ are adequate, see Lemmas \ref{lemma-pushforward-adequate} and \ref{lemma-base-change-adequate}. Hence $F$ as a kernel of a map of adequate functors is adequate, see Lemma \ref{lemma-kernel-adequate}.
Choose a set $\{L_i\}_{i \in I}$ of linearly adequate functors such that every linearly adequate functor is isomorphic to one of the $L_i$. This is possible. Suppose that we can find $Q(F) \to F$ with (1) and (2)' or every $i \in I$ the map $\Hom(L_i, Q(F)) \to \Hom(L_i, F)$ is a bijection. Then (2) holds. Namely, combining Lemmas \ref{lemma-adequate-surjection-from-linear} and \ref{lemma-kernel-adequate} we see that every adequate functor $G$ sits in an exact sequence $$ K \to L \to G \to 0 $$ with $K$ and $L$ direct sums of linearly adequate functors. Hence (2)' implies that $\Hom(L, Q(F)) \to \Hom(L, F)$ and $\Hom(K, Q(F)) \to \Hom(K, F)$ are bijections, whence the same thing for $G$. \medskip\noindent Consider the category $\mathcal{I}$ whose objects are pairs $(i, \varphi)$ where $i \in I$ and $\varphi : L_i \to F$ is a morphism. A morphism $(i, \varphi) \to (i', \varphi')$ is a map $\psi : L_i \to L_{i'}$ such that $\varphi' \circ \psi = \varphi$. Set $$ Q(F) = \colim_{(i, \varphi) \in \Ob(\mathcal{I})} L_i $$ There is a natural map $Q(F) \to F$, by Lemma \ref{lemma-colimit-adequate} it is adequate, and by construction it has property (2)'.
This lemma just collects some facts we have already seen so far. Part (1) is clear from the definitions, the characterization of weak Serre subcategories (see Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}), and Lemmas \ref{lemma-cokernel-adequate}, \ref{lemma-kernel-adequate}, and \ref{lemma-extension-adequate}. Recall that $\mathcal{P}$ is equivalent to the category $\textit{PMod}((\textit{Aff}/\Spec(A))_\tau, \mathcal{O})$. Hence (2) by Injectives, Proposition \ref{injectives-proposition-presheaves-modules}. Part (3) follows from Lemma \ref{lemma-adjoint} and Categories, Lemma \ref{categories-lemma-adjoint-exact}. Parts (4) and (5) follow from Homology, Lemmas \ref{homology-lemma-adjoint-preserve-injectives} and \ref{homology-lemma-adjoint-enough-injectives}.