category ${\mathscr{C}}_S$ has objects$${\text{Ob}}({\mathscr{C}}_S)=\{a'\,|\,a\in{\text{Ob}}({\mathscr{C}})\},$$and moduli spaces given by$$\begin{aligned}
{\mathcal{M}}(x',b')&=&{\mathcal{M}}(x,b)\sqcup{\mathcal{M}}(y,b),\\
{\mathcal{M}}(a',y')&=&{\mathcal{M}}(a,x)\sqcup{\mathcal{M}}(a,y),\end{aligned}$$In all other cases $${\mathcal{M}}(a',b') = {\mathcal{M}}(a,b) \sqcup \left({\mathcal{M}}(y,b) \times [0,1] \times {\mathcal{M}}(a,x)\right).$$

The main thing to note is that after we cancel $f$ against ${\overline}{x}$ in the intermediate flow category, we are left with object set $\{{\overline}{a}\,|\,a\in{\text{Ob}}({\mathscr{C}}){\setminus}\{x\}\}\cup\{e\}$. Hence the object $x'\in{\text{Ob}}({\mathscr{C}}_S)$ in the statement of the lemma is a relabelling of $e$. The rest of the lemma follows directly from [@JLSmorse Definition 2.9].

The following is a schematic for sliding $x$ over $y$, where the framings determined by a $(\pm)$–slide are also shown. The signs are calculated by following the construction through the handle cancellation in [@JLSmorse Definition 2.9].

$$\xymatrix @R=1cm @C=0.5cm {{\mathscr{C}}:&\ar[d]_-{\alpha}&&\ar[d]^-{\beta}&&&{\mathscr{C}}_S:&\ar[d]_-{\alpha}\ar[drr]^-{\mp\alpha}&&\ar[d]^-{\beta}\\
&\bullet_{x}\ar[d]_-{\gamma}&&\bullet_{y}\ar[d]^-{\delta}&&&&\bullet_{x'}\ar[d]_-{\gamma}\ar[drr]_-{\pm\delta}&&\bullet_{y'}\ar[d]^-{\delta}\\
&&&&&&&&&}$$

Now we are in a position to deduce the main result of this section.

Define the intermediate framed flow category $({\overline}{{\mathscr{C}}},{\overline}{\iota},{\overline}{{\varphi}}(\pm))$ as in Definition \[def:int\] and Proposition \[prop:immerse\]. By Proposition \[prop:choices\], the framed flow category obtained from cancelling $e$ against $f$ is precisely $({\mathscr{C}}, \iota,{\varphi})$. By sufficiently increasing ${\textbf{d}}$ to some ${\overline}{{\textbf{d}}}$ we may perturb the framed neat immersions $({\overline}{\iota},{\overline}{{\varphi}}(\pm))$, $(\iota_S,{\varphi}_S)$ to framed neat embeddings and hence geometrically realise the CW-complexes $|{\overline}{{\mathscr{C}}}|$ and $|{\mathscr{C}}_S|$. It is immediately clear from [@JLSmorse Theorem 2.17] that there are 