$l$. Then $$(M\times[0,1])^{(l)}=(M^{(l)}\times\{0,1\})\sqcup (M^{(l+1)}\times(0,1)).$$But $$\iota'(M^{(l)}\times\{0,1\})\subset ({\mathbb{E}}^{(d_k,\dots,d_{k+j})})^{(l)}\times\{0\}\times{\mathbb{R}},$$and $$\iota'(M^{(l+1)}\times(0,1))\subset ({\mathbb{E}}^{(d_k,\dots,d_{k+j})})^{(l+1)}\times({\mathbb{R}}_+\setminus\{0\})\times{\mathbb{R}},$$so that both components of $M\times[0,1])^{(l)}$ are embedded in $({\mathbb{E}}^{(d_k,\dots,d_{k+j})}\times{\mathbb{R}}_+\times{\mathbb{R}})^{(l)}$ as required. Moreover, as $\iota$ was a neat embedding, and the arc $\gamma_{u,v}$ meets the ${\mathbb{R}}$–axis transversely, $\iota'$ meets the boundary transversality condition (2) of Definition \[def:neat\] above.

The two different boundary framings ${\varphi}_1$, ${\varphi}_2$ will now be extended to framings of $\iota'$. To build the required framing ${\varphi}''_1$, take the embedding $${\varphi}':M\times[0,1]\times[-{\varepsilon},{\varepsilon}]^A\looparrowright {\mathbb{E}}^{\textbf{d}}\times{\mathbb{R}}_+\times{\mathbb{R}}$$ and extend it in a direction normal to $\gamma_{u,v}$ in ${\mathbb{R}}_+\times{\mathbb{R}}$ so that on the boundary we have ${\varphi}_1$ as required. To build the required framing ${\varphi}_2''$, we must look at the normal direction in ${\mathbb{E}}^{\textbf{d}}$ that was flipped by $R$ and turn that normal direction through $180^\circ$ as we move along $\gamma_{u,v}$. We do this by framing $\gamma_{u,v}\subset {\mathbb{R}}\times{\mathbb{R}}_+\times{\mathbb{R}}$ as follows. Referring to the coordinates of ${\mathbb{R}}\times{\mathbb{R}}_+\times{\mathbb{R}}$ as $e_1$, ${\overline}{e}$ and $e_2$, respectively, we will rotate $180^\circ$ around the $e_2$–axis, such that the first vector equals ${\overline}{e}$ halfway through (compare [@MR3252965 Lemma 3.1]). The rotation of the 2–dimensional frame as we move along $\gamma_{u,v}$ (extended trivially to the other normal directions) now defines the required embedding $${\varphi}_2'':M\times[0,1]\times[-{\varepsilon},{\varepsilon}]^{A+1}\looparrowright {\mathbb{E}}^{\textbf{d}}\times{\mathbb{R}}_+\times{\mathbb{R}}{\rm .}$$

We now add the objects $e$ and $f$ to 