Represent $K$ by a K-injective complex $\mathcal{J}^\bullet$ of sheaves of $\Lambda$-modules on $X'_\etale$. Choose a quasi-isomorphism $g'_!\mathcal{J}^\bullet \to \mathcal{I}^\bullet$ to a K-injective complex $\mathcal{I}^\bullet$ of sheaves of $\Lambda$-modules on $X_\etale$. The proof of Lemma \ref{lemma-shriek-proper-and-open} constructs $g_!Rf'_*K \to Rf_*g'_!K$ as $$ g_!f'_*\mathcal{J}^\bullet = g_!f'_!\mathcal{J}^\bullet = f_!g'_!\mathcal{J}^\bullet = f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet $$ Choose a quasi-isomorphism $(a')^{-1}\mathcal{J}^\bullet \to \mathcal{J}_1^\bullet$ to a K-injective complex $\mathcal{J}_1^\bullet$ of sheaves of $\Lambda$-modules on $X'_{1, \etale}$. Then we can pick a diagram of complexes $$ \xymatrix{ g'_{1, !}\mathcal{J}_1^\bullet \ar[rr] & & \mathcal{I}_1^\bullet \\ g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[u] \ar@{=}[r] & a^{-1}g'_!\mathcal{J}^\bullet \ar[r] & a^{-1}\mathcal{I}^\bullet \ar[u] } $$ commuting up to homotopy where all arrows are quasi-isomorphisms, the equality comes from Lemma \ref{lemma-proper-f-shriek}, and $\mathcal{I}_1^\bullet$ is a K-injective complex of sheaves of $\Lambda$-modules on $X_{1, \etale}$. The map $g_{1, !}Rf'_{1, *}(a')^{-1}K \to Rf_{1, *}g'_{1, !}(a')^{-1}K$ is given by $$ g_{1, !}f'_{1, *}\mathcal{J}_1^\bullet = g_{1, !}f'_{1, !}\mathcal{J}_1^\bullet = f_{1, !}g'_{1, !}\mathcal{J}_1^\bullet = f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet \to f_{1, *}\mathcal{I}_1^\bullet $$ The identifications across the $3$ equal signs in both arrows are compatible with pullback maps, i.e., the diagram $$ \xymatrix{ b^{-1}g_!f'_*\mathcal{J}^\bullet \ar@{=}[d] \ar[r] & g_{1, !}(b')^{-1}f'_*\mathcal{J}^\bullet \ar[r] & g_{1, !}f'_{1, *}(a')^{-1}\mathcal{J}^\bullet \ar@{=}[d] \\ b^{-1}f_*g'_!\mathcal{J}^\bullet \ar[r] & f_{1, *}a^{-1}g'_!\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet } $$ of complexes of abelian sheaves commutes. To show this it is enough to show the diagram commutes with $g_!, g_{1, !}, g'_!, g'_{1, !}$ replaced by $g_*, g_{1, *}, g'_*, g'_{1, *}$ (because the shriek functors are defined as subfunctors of the $*$ functors and the base change maps are defined in a manner compatible with this, see proof of Lemma \ref{lemma-base-change-f-shriek-separated}). For this new diagram the commutativity follows from the compatibility of pullback maps with horizontal and vertical stacking of diagrams, see Sites, Remarks \ref{sites-remark-compose-base-change} and \ref{sites-remark-compose-base-change-horizontal} so that going around the diagram in either direction is the pullback map for the base change of $f \circ g' = g \circ f'$ by $b$. Since of course $$ \xymatrix{ g_{1, !}f'_{1, *}(a')^{-1}\mathcal{J}^\bullet \ar@{=}[d] \ar[r] & g_{1, !}f'_{1, *}\mathcal{J}_1^\bullet \ar@{=}[d] \\ f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet  } $$ commutes, to finish the proof it suffices to show that $$ \xymatrix{ b^{-1}f_*g'_!\mathcal{J}^\bullet \ar[r] \ar[d] & f_{1, *}a^{-1}g'_!\mathcal{J}^\bullet \ar[r] \ar[d] & f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet \ar[d] \\ b^{-1}f_*\mathcal{I}^\bullet \ar[r] & f_{1, *}a^{-1}\mathcal{I}^\bullet \ar[rr] & & f_{1, *}\mathcal{I}_1^\bullet } $$ commutes in the derived category, which holds by our choice of maps earlier.