Note that assumption (1) implies that if $\{x_i \to x\}$ is a covering of $\mathcal{X}_\tau$ and $x \in \Ob(\mathcal{X}')$, then we have $x_i \in \Ob(\mathcal{X}')$. Hence we see that $\mathcal{X}' \to \mathcal{X}$ is continuous and cocontinuous as the coverings of objects of $\mathcal{X}'_\tau$ agree with their coverings seen as objects of $\mathcal{X}_\tau$. We obtain the morphism $g$ and the functor $g^{-1}$ is identified with the restriction functor, see Sites, Lemma \ref{sites-lemma-when-shriek}. \medskip\noindent In particular, if $\{x_i \to x\}$ is a covering in $\mathcal{X}'_\tau$, then for any abelian sheaf $\mathcal{F}$ on $\mathcal{X}$ then $$ \check H^p(\{x_i \to x\}, g^{-1}\mathcal{F}) = \check H^p(\{x_i \to x\}, \mathcal{F}) $$ Thus if $\mathcal{I}$ is an injective abelian sheaf on $\mathcal{X}_\tau$ then we see that the higher {\v C}ech cohomology groups are zero (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-injective-trivial-cech}). Hence $H^p(x, g^{-1}\mathcal{I}) = 0$ for all objects $x$ of $\mathcal{X}'$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}). In other words injective abelian sheaves on $\mathcal{X}_\tau$ are right acyclic for the functor $H^0(x, g^{-1}-)$. It follows that $H^p(x, g^{-1}\mathcal{F}) = H^p(x, \mathcal{F})$ for all $\mathcal{F} \in \textit{Ab}(\mathcal{X})$ and all $x \in \Ob(\mathcal{X}')$. \medskip\noindent Choose an object $x \in \mathcal{X}'$ lying over a scheme $U$ as in assumption (2). In particular $\mathcal{X}/x \to \mathcal{X}$ is a morphism of algebraic stacks which representable by algebraic spaces, surjective, and smooth. (Note that $\mathcal{X}/x$ is equivalent to $(\Sch/U)_{fppf}$, see Lemma \ref{lemma-localizing}.) The map of sheaves $$ h_x \longrightarrow * $$ in $\Sh(\mathcal{X}_\tau)$ is surjective. Namely, for any object $x'$ of $\mathcal{X}$ there exists a $\tau$-covering $\{x'_i \to x'\}$ such that there exist morphisms $x'_i \to x$, see Lemma \ref{lemma-surjective-flat-locally-finite-presentation}. Since $g$ is exact, the map of sheaves $$ g^{-1}h_x \longrightarrow * = g^{-1}* $$ in $\Sh(\mathcal{X}'_\tau)$ is surjective also. Let $h_{x, n}$ be the $(n + 1)$-fold product $h_x \times \ldots \times h_x$. Then we have spectral sequences \begin{equation} \label{equation-spectral-sequence-one} E_1^{p, q} = H^q(h_{x, p}, \mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F}) \end{equation} and \begin{equation} \label{equation-spectral-sequence-two} E_1^{p, q} = H^q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}'_\tau, g^{-1}\mathcal{F}) \end{equation} see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-to-cohomology-sheaf-sets}. \medskip\noindent Case I: $\mathcal{X}$ has a final object $x$ which is also an object of $\mathcal{X}'$. This case follows immediately from the discussion in the second paragraph above. \medskip\noindent Case II: $\mathcal{X}$ is representable by an algebraic space $F$. In this case the sheaves $h_{x, n}$ are representable by an object $x_n$ in $\mathcal{X}$. (Namely, if $\mathcal{S}_F = \mathcal{X}$ and $x : U \to F$ is the given object, then $h_{x, n}$ is representable by the object $U \times_F \ldots \times_F U \to F$ of $\mathcal{S}_F$.) It follows that $H^q(h_{x, p}, \mathcal{F}) = H^q(x_p, \mathcal{F})$. The morphisms $x_n \to x$ lie over smooth morphisms of schemes, hence $x_n \in \mathcal{X}'$ for all $n$. Hence $H^q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) = H^q(x_p, g^{-1}\mathcal{F})$. Thus in the two spectral sequences (\ref{equation-spectral-sequence-one}) and (\ref{equation-spectral-sequence-two}) above the $E_1^{p, q}$ terms agree by the discussion in the second paragraph. The lemma follows in Case II as well. \medskip\noindent Case III: $\mathcal{X}$ is an algebraic stack. We claim that in this case the cohomology groups $H^q(h_{x, p}, \mathcal{F})$ and $H^q(g^{-1}h_{x, n}, g^{-1}\mathcal{F})$ agree by Case II above. Once we have proved this the result will follow as before. \medskip\noindent Namely, consider the category $\mathcal{X}/h_{x, n}$, see Sites, Lemma \ref{sites-lemma-localize-topos-site}. Since $h_{x, n}$ is the $(n + 1)$-fold product of $h_x$ an object of this category is an $(n + 2)$-tuple $(y, s_0, \ldots, s_n)$ where $y$ is an object of $\mathcal{X}$ and each $s_i : y \to x$ is a morphism of $\mathcal{X}$. This is a category over $(\Sch/S)_{fppf}$. There is an equivalence $$ \mathcal{X}/h_{x, n} \longrightarrow (\Sch/U)_{fppf} \times_\mathcal{X} \ldots \times_\mathcal{X} (\Sch/U)_{fppf} =: \mathcal{U}_n $$ over $(\Sch/S)_{fppf}$. Namely, if $x : (\Sch/U)_{fppf} \to \mathcal{X}$ also denotes the $1$-morphism associated with $x$ and $p : \mathcal{X} \to (\Sch/S)_{fppf}$ the structure functor, then we can think of $(y, s_0, \ldots, s_n)$ as $(y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)$ where $y$ is an object of $\mathcal{X}$, $f_i : p(y) \to p(x)$ is a morphism of schemes, and $\alpha_i : y \to x(f_i)$ an isomorphism. The category of $2n+3$-tuples $(y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)$ is an incarnation of the $(n + 1)$-fold fibred product $\mathcal{U}_n$ of algebraic stacks displayed above, as we discussed in Section \ref{section-cech}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-on-sheaf-sets} we have $$ H^p(\mathcal{U}_n, \mathcal{F}|_{\mathcal{U}_n}) = H^p(\mathcal{X}/h_{x, n}, \mathcal{F}|_{\mathcal{X}/h_{x, n}}) = H^p(h_{x, n}, \mathcal{F}). $$ Finally, we discuss the ``primed'' analogue of this. Namely, $\mathcal{X}'/h_{x, n}$ corresponds, via the equivalence above to the full subcategory $\mathcal{U}'_n \subset \mathcal{U}_n$ consisting of those tuples $(y, f_0, \ldots, f_n, \alpha_0, \ldots, \alpha_n)$ with $y \in \mathcal{X}'$. Hence certainly property (1) of the statement of the lemma holds for the inclusion $\mathcal{U}'_n \subset \mathcal{U}_n$. To see property (2) choose an object $\xi = (y, s_0, \ldots, s_n)$ which lies over a scheme $W$ such that $(\Sch/W)_{fppf} \to \mathcal{U}_n$ is smooth and surjective (this is possible as $\mathcal{U}_n$ is an algebraic stack). Then $(\Sch/W)_{fppf} \to \mathcal{U}_n \to (\Sch/U)_{fppf}$ is smooth as a composition of base changes of the morphism $x : (\Sch/U)_{fppf} \to \mathcal{X}$, see Algebraic Stacks, Lemmas \ref{algebraic-lemma-base-change-representable-transformations-property} and \ref{algebraic-lemma-composition-representable-transformations-property}. Thus axiom (1) for $\mathcal{X}$ implies that $y$ is an object of $\mathcal{X}'$ whence $\xi$ is an object of $\mathcal{U}'_n$. Using again $$ H^p(\mathcal{U}'_n, \mathcal{F}|_{\mathcal{U}'_n}) = H^p(\mathcal{X}'/h_{x, n}, \mathcal{F}|_{\mathcal{X}'/h_{x, n}}) = H^p(g^{-1}h_{x, n}, g^{-1}\mathcal{F}). $$ we now can use Case II for  $\mathcal{U}'_n \subset \mathcal{U}_n$ to conclude.