The primary obstruction cochain is a cocycle

First suppose $n\ge 2$. Work in one component and in a supplied universal-cover coordinate system. The lift of $X^n\cup A$ is simply connected: adjoining the remaining relative cells, whose dimensions are at least three, does not change ${\pi }_1$. Hence [F1] identifies each lifted $(n+1)$-cell characteristic class with its oriented relative cellular generator.

Now suppose $n=1$. Put $B=X^1\cup A$ and $W=X^2\cup A$. On each component $B_i$ with its supplied basepoint and whisker, $f_{\ast }:{\pi }_1(B_i)\to {\pi }_1(Y,f(b_i))$ has abelian target by hypothesis. Thus [F4] gives a unique homomorphism ${\lambda }_i:H_1(B_i;\mathbb{Z})\to {\pi }_1(Y,f(b_i))$ with $f_{\ast }={\lambda }_i\circ h_{B_i}$. For an oriented relative two-cell $e$, let $u_e\in H_2(W,B;\mathbb{Z})$ be its characteristic disk class. Its pair boundary $\mathrm{\partial }{u}_e\in H_1(B_i;\mathbb{Z})$ is the Hurewicz class of the attaching loop, including the supplied orientation and whisker. Therefore $\theta (f)(e)={\lambda }_i(\mathrm{\partial }{u}_e)$. The assumed trivial conjugation action makes this formula independent of loop transport in the target and makes the $n=1$ coefficient system constant in these component coordinates. No representatives are selected simultaneously.

In this $n\ge 2$ case, the geometric obstruction on a lifted $(n+1)$-cell is the value on its cellular generator of the composite “inverse relative Hurewicz, relative boundary, then $f_{\ast }$,” with the prescribed whisker transport. The coordinate rule is equivariant under deck transformations, so it descends to the local cochain $\theta (f)$ of the definition.

For $n=1$, let $c$ be an oriented relative three-cell. Its attaching sphere determines $v_c\in H_2(W;\mathbb{Z})$; under the relative inclusion $i_{\ast }:H_2(W;\mathbb{Z})\to H_2(W,B;\mathbb{Z})$, the class $i_{\ast }{v}_c$ is the relative cellular boundary of $c$, by the connecting-map and signed-incidence description in [F3]. The sphere and all its boundary incidences lie in one component, so Step 1.2 gives $(\delta \theta (f))(c)={\lambda }_i\left(\mathrm{\partial }{i}_{\ast }{v}_c\right)=0.$ The last equality is exactness of the pair homology sequence [F5]. This argument retains every cell and path in the arbitrary subcomplex $A$ inside the pair $(W,B)$; it never assumes that $A$ is simply connected.

For $n\ge 2$, let $c$ be an oriented lifted $(n+2)$-cell. Naturality of relative Hurewicz for the two consecutive skeletal pairs identifies the cellular boundary of $c$ with the Hurewicz image of its attaching class in ${\pi }_{n+1}(X^{n+1}\cup A,X^n\cup A)$. Evaluating $\theta (f)$ on that boundary is therefore $f_{\ast }$ applied after the next relative boundary. The consecutive maps ${\pi }_{n+1}(X^{n+1}\cup A)\to {\pi }_{n+1}(X^{n+1}\cup A,X^n\cup A)\to {\pi }_n(X^n\cup A)$ have zero composite by [F2]. Thus $(\delta \theta (f))(c)=0$.

The relative $(n+2)$-cells freely generate the cellular chain module, so Steps 3.1 and 2.2 give $\delta \theta (f)=0$ in their respective ranges, component by component. For $n\ge 2$, [F3] includes exactly the whisker monodromy used in Step 2.1; for $n=1$, it is the identity by Step 1.2. Hence $\theta (f)$ defines a cellular cohomology class, and the AT-23 comparison in [F3] carries it naturally to the stated singular local-coefficient class. No simultaneous choices beyond the supplied coordinates are made. $\mathrm{\square }$