Serre edge maps come from projection and fiber inclusion

Apply the Serre construction to the fibration $F_b\to \{b\}$. Its second page is concentrated in the column $a=0$, with $E_{0,n}^2=H_n(F_b;R)$; its image filtration has $F_0{H}_n=H_n(F_b;R)$. Hence its fiber edge is the identity under these canonical identifications. The square from this fibration to $p$, with total map $i_b$ and cellular base inclusion $\{b\}\hookrightarrow B$, has second-page vertical-axis map ${\kappa }_b$. Naturality in [F2] therefore gives ${ϵ}_F{\kappa }_b=(i_b{)}_{\ast }$.

Map $p$ to the identity fibration $1_B:B\to B$ by the square with total map $p$ and base map $1_B$. On a fiber this is the collapse $F_x\to \{x\}$, whose map on $H_0$ is the augmentation. Thus the second-page bottom-row map is $a_{\ast }$. The identity fibration has second page concentrated in row zero and its base edge is the identity on $H_n(B;R)$. Edge naturality in [F2] gives $a_{\ast }{ϵ}_B=p_{\ast }$.

To see directly that ${\kappa }_b$ is epic, [F3] presents $H_0(B;{\mathcal{H}}_n)$ as the direct sum of all stalks modulo the relations $T_{\gamma }(m)$ at the terminal vertex minus $m$ at the initial vertex. For any generator $m$ in a stalk at $x$, path connectedness supplies a path from $b$ to $x$, and its relation expresses $m$ as the image under ${\kappa }_b$ of the inverse transport of $m$. This is a one-generator argument and makes no simultaneous selection of paths. The same relation and the homotopy in $E$ traced by fiber transport show directly that $(i_b{)}_{\ast }$ kills the kernel relations, agreeing with the naturality proof in Step 1.1.

If every fiber is nonempty and path connected, its augmentation $H_0(F_x;R)\to R$ is an isomorphism, and fiber transport commutes with augmentation. Its inverse sends $1$ to the class of any point; path connectedness makes that class independent of the point, so the inverse is canonical and natural in $x$. Hence $a_{\ast }$ is the canonical identification with ordinary coefficients from [F3], and Step 1.2 identifies ${ϵ}_B$ itself with $p_{\ast }$.

For $n=0$, the degree-zero local boundary relations used in Step 2.1 are exactly the relevant coinvariant relations. Empty fibers give zero stalks and a zero fiber edge; if the base is empty there is no supplied $b$ and only the zero base-edge assertion remains. The zero ring, point fibers, the identity fibration, constant monodromy, one path relation, constant paths, and degenerate simplices are all covered by Steps 1.1–2.2. Both edge directions and both factorization equalities are explicit. Each path is chosen only for one displayed generator, so no AC is used. The proposition has no iff assertion.