Actual analytic constructible sheaves admit coherent six operations

The operations restrict to $D_c^b$ and satisfy the stated adjunctions and diagrams. The compact-support adjoint and the coherent six-operation suppliers [F1] construct the functors together with their adjunction units and counits, not merely unrelated isomorphisms: their composition, Cartesian exchange, projection, trace and localization diagrams are the asserted diagrams, including proper equality and the smooth local-product comparisons. The finite conical and incidence suppliers [F2] prove bounded constructibility and finite coefficient models for the stated finite collection, so the constructed operations restrict to $D_c^b$; and [F3] supplies the right adjoint $R{f}_{!}⊣f^{!}$ on that specialization. All vector-space and injective constructions apply over an arbitrary field: the signed cochains use only $0$ and the units $\pm 1$, so no characteristic hypothesis enters.

Proper equality and involutive duality. Proper $f$ satisfies $R{f}_{!}=R{f}_{\ast }$ by [F1]. The oriented stratified Verdier supplier [F4] gives the evaluation isomorphism and proves the natural evaluation $A\to D_X{D}_{X}A$ is an isomorphism on bounded constructible objects, so duality is involutive there; its adjunction construction gives the exchange of $R{f}_{\ast }$, $R{f}_{!}$ with the duality functors, and the extraordinary proper base change. All these maps use the same ordinary units, counits and orientation, so their coherence is the coherence of the actual construction and is compatible with the pasting and trace identities of [F1].

Smooth normalization and relative pullback. On a smooth complex $d$-fold the complex orientation identifies ${\omega }_X\simeq k_X[2d]$ by [F4]. Let $a:X\to Y$ be smooth of pure relative complex dimension $d$. Smooth analytic local charts identify it, locally on source and target, with $\mathrm{pr}:W\times V\to V$, where $W$ is an oriented complex $d$-ball; $V$ need not be smooth. For an open $V^{\mathrm{\prime }}\subseteq V$, the generator $E_{W\times V^{\mathrm{\prime }}}=j_{W\times V^{\mathrm{\prime }}!}k$ has proper-support image $E_{V^{\mathrm{\prime }}}{\otimes }_{k}R{\mathrm{\Gamma }}_c(W,k)=E_{V^{\mathrm{\prime }}}[-2d]$, by the finite oriented ball relative-cochain model of [F2] and the support construction of [F1]. For any bounded constructible $B$ on $V$, the ordinary product comparison of [F2] gives $R\mathrm{\Gamma }(W\times V^{\mathrm{\prime }},{\mathrm{pr}}^{\ast }B)=R\mathrm{\Gamma }(V^{\mathrm{\prime }},B)$ because $W$ is contractible. Consequently $R\mathrm{Hom}(E_{W\times V^{\mathrm{\prime }}},{\mathrm{pr}}^{\ast }B[2d])=R\mathrm{Hom}(R{\mathrm{pr}}_{!}{E}_{W\times V^{\mathrm{\prime }}},B)$. The orientation integration map $R{\mathrm{pr}}_{!}{\mathrm{pr}}^{\ast }B[2d]\to B$ transposes under [F3] to a comparison ${\mathrm{pr}}^{\ast }B[2d]\to {\mathrm{pr}}^{!}B$, and on these generators its Hom map is exactly the displayed cochain evaluation isomorphism, natural under shrinking product charts; the exceptional adjoint of [F3] identifies the same right-hand complex with $R\mathrm{Hom}(E_{W\times V^{\mathrm{\prime }}},{\mathrm{pr}}^{!}B)$. Product opens form a neighbourhood basis, so the comparisons identify derived sections and hence cohomology stalks, proving ${\mathrm{pr}}^{!}B\simeq {\mathrm{pr}}^{\ast }B[2d]$. The positive complex orientation makes the local comparisons agree on overlaps, so they glue to $a^{!}\simeq a^{\ast }[2d]$ on all bounded constructible inputs. The orientation generator maps to $1\in k$ over every field, including characteristic two. This proves the relative formula on arbitrary constructible complexes rather than only on smooth local systems.

Coefficient extension. The finite local cellular and cochain models of [F2] have finite-dimensional terms and $k$-linear comparison maps, so extending scalars along $k\to K$ is exact, commutes with finite sums, cohomology and finite-dimensional duals, and carries the models of [F1] and [F4] to the corresponding models of the extended coefficient field; the exceptional comparison follows from $f^{!}=D_X{f}^{\ast }{D}_Y$, and composition and traces extend because their maps are the same finite evaluation and section comparisons. This does not assert that infinite products of injective stalks commute with scalar extension.