Difference cochain between two cellular extensions

Retain the abelian coefficient hypotheses of the primary obstruction. Let $f_0,f_1:X^n\cup A\to Y$ and supply a homotopy

$H:(X^{n-1}\cup A)\times I⟶Y$

from their restrictions. Give every prism $e^n\times I$ the product orientation with the interval last, so

$\mathrm{\partial }(e^n\times I)=(\mathrm{\partial }{e}^n)\times I+(-1{)}^n(e^n\times \{1\}-e^n\times \{0\})\mathrm{.}$

The maps $f_0$ on the bottom, $H$ on the side, and $f_1$ on the top define a map from the boundary sphere of each prism to $Y$. Transport its homotopy class to the $f_0^{\ast }{\mathrm{\Pi }}_{n}Y$ coordinate using the basepoint track of $H$. With the sign convention of Davis--Kirk, define

$d(f_0,H,f_1)(e^n)=(-1{)}^{n+1}[f_0\cup H\cup f_1{]}_{\mathrm{\partial }(e^n\times I)}\mathrm{.}$

These values form the difference cochain

$d(f_0,H,f_1)\in C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^n(X,A;f_0^{\ast }{\mathrm{\Pi }}_{n}Y)\mathrm{.}$

The homotopy $H$ supplies the canonical natural identification between the coefficient systems of $f_0$ and $f_1$. The orientation sign is chosen so that the next theorem has the exact formula $\delta d=\theta (f_0)-\theta (f_1)$.