must describe the required modifications to work on the case $({\overline}{\iota},{\overline}{{\varphi}}(-))$. Again we will suppress the “$-$” notation as we work on this case.

The first difference from the $({\overline}{\iota},{\overline}{{\varphi}}(+))$ case is that we declare ${\overline}{{\varphi}}_{f,{\overline}{x}}$ be a $+$ framing and both of ${\overline}{{\varphi}}_{f,{\overline}{y}}$ and ${\overline}{{\varphi}}_{f,e}$ to be a $-$ framings. This, in turn, means that for each object $b$, we will want to use case (2) of Lemma \[lem:arc\] when we come to embed and frame the ${\mathcal{M}}(y,b)\times[0,1]$ component of ${\mathcal{M}}(f,{\overline}{b})$. In order to get into case (2) of this lemma, we must endow the ${\mathcal{M}}(y,b)$ component of ${\mathcal{M}}(e,{\overline}{b})$ with the framing obtained by taking the image of ${\varphi}_{y,b}$ in ${\mathbb{E}}^{\textbf{d}}[|y|:|b|]$ and performing the reflection $R$ in the plane through the origin and perpendicular to the first coordinate axis of the ${\mathbb{R}}^{d_{i-1}}$ factor. Other than these modifications, the construction of the immersions and framings in the $({\overline}{\iota},{\overline}{{\varphi}}(-))$ case proceeds exactly as the $({\overline}{\iota},{\overline}{{\varphi}}(+))$ case.

The $({\overline}{\iota},{\overline}{{\varphi}}(\pm))$ cases are illustrated in the following diagram, where $\alpha$, $\beta$, $\gamma$, $\delta$ denote framings and $-\delta$ represents the effect of reflecting $\delta$ by $R$ as described above.

$$\xymatrix @R=1cm @C=0.5cm {{\mathscr{C}}:&\ar[d]_-{\alpha}&&\ar[d]^-{\beta}&&&{\overline}{{\mathscr{C}}}:&\ar[d]_-\alpha&&\bullet_f\ar[d]^-{-}\ar[drr]^-{\pm}\ar[dll]_-{+}&&\ar[d]^-{\beta}\\
&\bullet_x\ar[d]_-{\gamma}&&\bullet_y\ar[d]^-{\delta}&&&&\bullet_{{\overline}{x}}\ar[d]_-{\gamma}&&\bullet_e\ar[drr]_-{\pm\delta}\ar[dll]^-{\gamma}&&\bullet_{{\overline}{y}}\ar[d]^-{\delta}\\
&&&&&&&&&&&}$$

\[prop:choices\]Cancelling $e$ against $f$ in $({\overline}{{\mathscr{C}}},{\overline}{\iota},{\overline}{{\varphi}}(\pm))$, using the handle cancellation 