(\sigma , \sigma_0)* \sigma \label{cohid2eqn}\end{aligned}$$ There is a natural isomorphism $$\begin{aligned}
(S\times [0,1])^{ \xi \times [0,1]} (\sigma , \sigma_0) & {\longrightarrow}{\operatorname{Hom}}( \sigma , \sigma_0) \\
 \mu & {\longmapsto}- {p_S}_* \mu \end{aligned}$$ using the projection $p_S:S \times [0,1]{\longrightarrow}S$. It induces an isomorphism between the coends in and and gives us the desired isomorphism .

To specify the natural isomorphism for 1-morphisms $(\Sigmait_a,\varphi_a) : (S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ and $(\Sigmait_b,\varphi_b) : (S_1,\xi_1) {\longrightarrow}(S_2,\xi_2)$ in ${T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$, we note that for a generator $\sigma_0 \in {\operatorname{Fund}}(S_0)$ $$\begin{aligned}
(T_\theta(\Sigmait_b,\varphi_b) \circ T_\theta(\Sigmait_a,\varphi_a)) \sigma_0 &= \int^{(\sigma_1,\sigma_2) \in {\operatorname{Fund}}(S_1)\times {\operatorname{Fund}}(S_2)}  \Sigmait^{\varphi_a}_a{\left(  {\sigma_1},{\sigma_0} \right)} \otimes \Sigmait_b^{\varphi_b}{\left(  {\sigma_2},{\sigma_1}\right)}  * \sigma_2 \  \\ &\cong \int^{\sigma_2 \in {\operatorname{Fund}}(S_2)} \left(\int^{\sigma_1 \in {\operatorname{Fund}}(S_1)}  \Sigmait^{\varphi_a}_a {\left(  {\sigma_1},{\sigma_0} \right)} \otimes \Sigmait_b^{\varphi_b}{\left(  {\sigma_2},{\sigma_1}\right)} \right) * \sigma_2,\label{eqnrefinnercoend}\end{aligned}$$ where we have used Fubini’s Theorem for coends. The action of a morphism $\lambda_1 :  \sigma_1 {\longrightarrow}\sigma'_1 \in {\operatorname{Fund}}(S_1)$ in the covariant component is induced by $$\begin{aligned}
\Sigmait_b^{\varphi_b}{\left( \sigma_2, \sigma_1\right)} & {\longrightarrow}\Sigmait_b^{\varphi_b} {\left( \sigma_2,\sigma'_1\right)} \\
\mu & {\longmapsto}\mu - \lambda_1.\end{aligned}$$ To compute the inner coend in we observe that $$\begin{aligned}
\Phiit_{\sigma_1,\sigma_2} :  \Sigmait_a^{\varphi_a} {\left( \sigma_0,\sigma_1\right)}\otimes \Sigmait_b^{\varphi_b}{\left( \sigma_1,\sigma_2\right)} & {\longrightarrow}(\Sigmait_b \circ \Sigmait_a)^{\varphi_b \cup \varphi_a} {\left(  \sigma_2,\sigma_0\right)}  \\
\mu_1\otimes \mu_2 & {\longmapsto}\mu_1 +\mu_2 \ .\end{aligned}$$ is a canonical isomorphism by Lemma \[lemmamscgluingcyclescob\]. Here $\varphi_b \cup 