A compatible presheaf-complex model of étale derived sections

Choose the big site $\mathcal{B}$ and the stated abelian resolution $a:A[0]\to I^{\bullet }$ using [A1], [F1] and [F3]. Restriction of $I^{\bullet }$ to ${\mathcal{C}}_F$ is the strict presheaf complex $C_{\mathrm{\Lambda }}$. For each $X$, $r_X{I}^{\bullet }$ is an abelian K-injective resolution of ${\underset{‾}{\mathrm{\Lambda }}}_X$, so its sections compute abelian derived sections. To model coefficients on this same big site, write $L={\underset{‾}{\mathrm{\Lambda }}}_{\mathcal{B}}$ and let $U$ forget its action. A sheaf of modules over the constant ring sheaf is equivalently an abelian sheaf with endomorphisms ${\rho }_{\lambda }$ for $\lambda \in \mathrm{\Lambda }$ satisfying ${\rho }_1=\mathrm{i}\mathrm{d}$, ${\rho }_{\lambda +\mu }={\rho }_{\lambda }+{\rho }_{\mu }$ and ${\rho }_{\lambda \mu }={\rho }_{\lambda }{\rho }_{\mu }$: scalar sections are locally represented by constant scalars, so the action extends uniquely by sheaf gluing. This description applies on both big and small sites. Lift ${\mathrm{\ell }}_X$ from [F6] by replacing each ${\rho }_{\lambda }$ by ${\mathrm{\ell }}_X({\rho }_{\lambda })$. Additivity and functoriality preserve the action identities. Naturality of the adjunction bijection in both variables shows that a map commutes with every scalar if and only if its adjunct does; hence the lifted functor is left adjoint to coefficient restriction. Kernels and cokernels inherit scalar actions, and exactness is detected after forgetting them, so this lifted left adjoint remains exact. It follows directly that coefficient restriction preserves K-injectivity: homotopy classes of maps from an acyclic coefficient complex $T$ into $r_{X}J$ are the classes of maps from ${\mathrm{\ell }}_{X}T$ into $J$, which vanish for K-injective $J$ because the exact ${\mathrm{\ell }}_X$ makes ${\mathrm{\ell }}_{X}T$ acyclic. Choose a module K-injective resolution $b:L[0]\to J^{\bullet }$ by [F7]. Thus $r_X{J}^{\bullet }$ is a coefficient K-injective resolution on $X_{\mathrm{e}\mathrm{t}}$, and the strict coefficient presheaf complex $P_{\mathrm{\Lambda }}(X)=J^{\bullet }(X)$ computes the coefficient derived sections, with the action restricted along $\mathrm{\Lambda }\to L(X)$. For $f:Y\to X$, apply [F4] to $b$: exact inverse image and restriction, together with the constant-object isomorphism, make ${\beta }_f:f_{small}^{-1}{r}_X{J}^{\bullet }\to r_Y{J}^{\bullet }$ a quasi-isomorphism. The induced map on sections is exactly $J^{\bullet }(f)$, and is the actual derived pullback of [F2]. This is a simultaneous resolution of the entire constant-object diagram: its coaugmentation is compatible with every ${\beta }_f$, its component targets are K-injective, and the transition identity holds for every composable string. In the enhanced resolution computation of [F2], applying inverse image followed by these replacement maps therefore computes that same coherent functor. The associativity identifications come from iterated fibre products; their universal properties make all iterated identifications agree. Consequently this comparison retains the coherence for all lengths of strings before localization, rather than only the individual derived-category arrows. The localized $P_{\mathrm{\Lambda }}$ is thus the actual coefficient-valued $E_{\mathrm{\Lambda }}$. This argument concerns constant-ring actions and does not apply the false structure-sheaf-module analogue of [F3].

Construct the coefficient-forgetting comparison on $\mathcal{B}$ before any scheme evaluation. Choose an abelian K-injective replacement $q:U{J}^{\bullet }\to K^{\bullet }$ by [F1], and put $c=qU(b):A[0]\to K^{\bullet }$. Reference 2 applied on the big ringed site $(\mathcal{B},L)$ to $J^{\bullet }$ and $q$ gives a quasi-isomorphism $q(X):U{J}^{\bullet }(X)\to K^{\bullet }(X)$ for every object $X$ of $\mathcal{B}$. Both $a$ and $c$ are sheaf quasi-isomorphisms. The Hom bijection of [F7], applied to precomposition with $a$ and target $K^{\bullet }$, gives a global chain map $h:I^{\bullet }\to K^{\bullet }$ and a global sheaf cochain homotopy $ha\simeq c$. Therefore $h$ is a quasi-isomorphism; since $I^{\bullet }$ and $K^{\bullet }$ are K-injective it is a cochain-homotopy equivalence by [F7]. Evaluation preserves the inverse maps and homotopies, so $h(X)$ is a quasi-isomorphism for every $X$. We have the diagram-level zigzag $U{P}_{\mathrm{\Lambda }}\stackrel{q(-)}{\to }{K}^{\bullet }(-)\stackrel{h(-)}{\leftarrow }{C}_{\mathrm{\Lambda }}$ of objectwise quasi-isomorphisms of strict presheaf complexes. For $f:Y\to X$, strict naturality gives $K^{\bullet }(f)q(X)=q(Y)U{J}^{\bullet }(f)$ and $K^{\bullet }(f)h(X)=h(Y)I^{\bullet }(f)$. Before taking sections, naturality of [F4] gives $(r_{Y}q){\beta }_f^{UJ}={\beta }_f^K{f}_{small}^{-1}(r_{X}q)$ and the analogous identity for $h$; the one sheaf homotopy $ha\simeq c$ restricts compatibly to every object as well. Hence both zigzag arrows respect all inverse-image transition maps, coaugmentations up to that specified global homotopy, and every iterated composite simultaneously. Taking nerves and then enhanced localization turns the strict zigzag into natural transformations of the full coherent diagrams, with all components equivalences. Inverting its right arrow yields the required coherent equivalence from the underlying coefficient functor of step 1.1 to $C_{\mathrm{\Lambda }}$, inducing the identity on the constant object. This constructs the coefficient-forgetting comparison rather than inferring its enhanced naturality from reference 2. Equivalences of diagrams preserve their limits by the universal property in [F5], so the same comparison applies to hypercover limits and augmentations. Degreewise products give the formal-family values by [F7]. Precomposition with the representative functor of [F1] gives the specified comparison outside the skeleton. All these constructions precede cdh sheafification.