technique of [@JLSmorse 2.3], results in precisely $({\mathscr{C}}, \iota, {\varphi})$.

Write $({\mathscr{C}}_H,\iota_H,{\varphi}_H)$ for the result of cancelling $e$ against $f$ as described in [@JLSmorse 2.3]. According to [@JLSmorse Definition 2.9], the objects of ${\mathscr{C}}$ and of ${\mathscr{C}}_H$ are in 1:1 correspondence. Moreover, recall from [@JLSmorse 2.3] that if $a,b$ are objects of ${\mathscr{C}}$, then the moduli spaces of ${\mathscr{C}}_H$ are given by$${\mathcal{M}}(a,b)={\mathcal{M}}({\overline}{a},{\overline}{b})\cup_f\left({\mathcal{M}}(f,{\overline}{b})\times {\mathcal{M}}({\overline}{a},e)\right),$$where $f$ is a certain identification described generally in [@JLSmorse 2.3]. However, when $a$ is an object of ${\mathscr{C}}$, there are no non-empty moduli spaces ${\mathcal{M}}({\overline}{a},e)$, so in fact the moduli spaces of ${\mathscr{C}}_H$ are the same as those of ${\mathscr{C}}$. This simplification, carried through the construction in [@JLSmorse Theorem 2.17], actually results in a precise agreement of embeddings and framings so that in fact $({\mathscr{C}},\iota,{\varphi})=({\mathscr{C}}_H,\iota_H,{\varphi}_H)$.

Handle sliding in framed flow categories {#subsec:sliding}
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The framed flow category $({\mathscr{C}}_S,\iota_S,{\varphi}_S(\pm))$, called the *effect of $(\pm)$–sliding $x$ over $y$ in $({\mathscr{C}},\iota,{\varphi})$* is defined as the result of cancelling $f$ against ${\overline}{x}$ in $({\overline}{{\mathscr{C}}},{\overline}{\iota},{\overline}{{\varphi}}(\pm))$. We will often suppress the “$+$” and “$-$” from the notation.

The following lemma is a straightforward consequence of the constructions so far and the definition of handle cancellation given in [@JLSmorse Definition 2.9].

The framed flow 