The orientation system is a local system

Assign $x\mapsto {\mathcal{O}}_x$ and $[\gamma :x\to y]\mapsto T_{\gamma }$. Homotopy invariance in [F1] makes the arrow map well defined. The constant-path and concatenation identities say respectively that identities and composition in ${\mathrm{\Pi }}_1(M)$ are preserved, while reversal says every transport is an isomorphism. This is precisely the functor required by [F2].

Put $({\mathcal{O}}_M^R{)}_x={\mathcal{O}}_x{\otimes }_{\mathbb{Z}}R$. On elementary tensors define $q\cdot (o\otimes r):=o\otimes (qr)\mathrm{.}$ The additive and balancing relations in [F3] are preserved by this formula; distributivity, associativity, and the unit law in $R$ therefore make the tensor product a left $R$-module. For a path $\gamma :x\to y$, the formula $o\otimes r⟼T_{\gamma }(o)\otimes r$ also preserves every relation in [F3], because $T_{\gamma }$ is an additive, hence $\mathbb{Z}$-linear, map. It descends to an $R$-linear map denoted $T_{\gamma }\otimes 1_R$. Identity and composition agree on elementary tensors, which generate the tensor product, and $T_{\bar{\gamma }}\otimes 1_R$ is the inverse. Thus these maps define an $R$-module local system. Finally, if $o_x$ generates ${\mathcal{O}}_x$, the maps $R⟶({\mathcal{O}}_M^R{)}_x,r⟼o_x\otimes r,(m{o}_x)\otimes r⟼mr$ are well defined inverse $R$-linear maps by the same relations. Hence every stalk is free of rank one, including the zero ring under the usual rank-one convention $R\cong R$.

Fix $x$ and a generator $o_x$. A loop transport is an automorphism of the infinite cyclic group ${\mathcal{O}}_x$, so $T_{\gamma }(o_x)=w_M(\gamma )o_x$ for a unique sign. Composition in step 1.1 makes $w_M$ a homomorphism. It is $+1$ exactly when the lifted path in the orientation cover returns to the chosen generator sheet, and $-1$ exactly when it returns to the other sheet, so it is the orientation character. Scalar extension gives the same multiplication by $w_M(\gamma )$ on ${\mathcal{O}}_x^R$. Changing $o_x$ to $-o_x$ does not change this sign.

For an orientable component, a continuous choice of generator identifies the system with the constant rank-one system; conversely such a trivialization selects one sheet of the orientation cover and orients the component. The empty manifold gives the empty functor, dimension zero gives constant stalks on a discrete space, and disconnected manifolds are handled componentwise. No global generator or path family is selected in the construction, so no AC is used.