In the terminology introduced above, this means that cohomology commutes with base change for every proper morphism of schemes. By Lemma \ref{lemma-reduce-to-P1} it suffices to prove that cohomology commutes with base change for the morphism $\mathbf{P}^1_S \to S$ for every scheme $S$. \medskip\noindent Let $S$ be the spectrum of a strictly henselian local ring with closed point $s$. Set $X = \mathbf{P}^1_S$ and $X_0 = X_s = \mathbf{P}^1_s$. Let $\mathcal{F}$ be a sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules on $X_\etale$. The key to our proof is that $$ H^q_\etale(X, \mathcal{F}) = H^q_\etale(X_0, \mathcal{F}|_{X_0}). $$ Namely, choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet$ by injective sheaves of $\mathbf{Z}/\ell\mathbf{Z}$-modules. Then $\mathcal{I}^\bullet|_{X_0}$ is a resolution of $\mathcal{F}|_{X_0}$ by right $H^0_\etale(X_0, -)$-acyclic objects, see Lemma \ref{lemma-efface-cohomology-on-fibre-by-finite-cover}. Leray's acyclicity lemma tells us the right hand side is computed by the complex $H^0_\etale(X_0, \mathcal{I}^\bullet|_{X_0})$ which is equal to $H^0_\etale(X, \mathcal{I}^\bullet)$ by Lemma \ref{lemma-h0-proper-over-henselian-local}. This complex computes the left hand side. \medskip\noindent Assume $S$ is general and $\mathcal{F}$ is a sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules on $X_\etale$. Let $\overline{s} : \Spec(k) \to S$ be a geometric point of $S$ lying over $s \in S$. We have $$ (R^qf_*\mathcal{F})_{\overline{s}} = H^q_\etale(\mathbf{P}^1_{\mathcal{O}_{S, \overline{s}}^{sh}}, \mathcal{F}|_{\mathbf{P}^1_{\mathcal{O}_{S, \overline{s}}^{sh}}}) = H^q_\etale(\mathbf{P}^1_{\kappa(s)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa(s)^{sep}}}) $$ where $\kappa(s)^{sep}$ is the residue field of $\mathcal{O}_{S, \overline{s}}^{sh}$, i.e., the separable algebraic closure of $\kappa(s)$ in $k$. The first equality by Theorem \ref{theorem-higher-direct-images} and the second equality by the displayed formula in the previous paragraph. \medskip\noindent Finally, consider any morphism of schemes $g : T \to S$ where $S$ and $\mathcal{F}$ are as above. Set $f' : \mathbf{P}^1_T \to T$ the projection and let $g' : \mathbf{P}^1_T \to \mathbf{P}^1_S$ the morphism induced by $g$. Consider the base change map $$ g^{-1}R^qf_*\mathcal{F} \longrightarrow R^qf'_*(g')^{-1}\mathcal{F} $$ Let $\overline{t}$ be a geometric point of $T$ with image $\overline{s} = g(\overline{t})$. By our discussion above the map on stalks at $\overline{t}$ is the map $$ H^q_\etale(\mathbf{P}^1_{\kappa(s)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa(s)^{sep}}}) \longrightarrow H^q_\etale(\mathbf{P}^1_{\kappa(t)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa(t)^{sep}}}) $$ Since $\kappa(s)^{sep} \subset \kappa(t)^{sep}$ this map is an isomorphism by Lemma \ref{lemma-base-change-dim-1-separably-closed}. \medskip\noindent This proves cohomology commutes with base change for $\mathbf{P}^1_S \to S$ and sheaves of $\mathbf{Z}/\ell\mathbf{Z}$-modules. In particular, for an injective sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules the higher direct images of any base change are zero. In other words, condition (2) of Lemma \ref{lemma-proper-base-change-in-terms-of-injectives} holds and the proof is complete.