\colon S_1 {\longrightarrow}S_2$ be 1-morphisms in ${{\mathbf{Cob}}}(n,n-1,n-2)$, and let $\nu \in C_{n-1}(\Sigmait_a)$ and $\nu' \in C_{n-1}(\Sigmait_b)$ be fundamental cycles with $\partial \nu =  \sigma_1 -  \sigma_0$ and $\partial \nu' =  \sigma_2 -  \sigma_1$ for fixed fundamental cycles $\sigma_j \in Z_{n-2}(S_j)$, $j=0,1,2$. Then by $\nu'\circ \nu:= \nu'+\nu \in  C_{n-1}(\Sigmait_b \circ \Sigmait_a) $ we get a fundamental cycle of $\Sigmait_b \circ \Sigmait_a$ satisfying $\partial (\nu'\circ \nu) =  \sigma_2 -  \sigma_0$.

\[lemmacompof1mor\] $T_\theta$ respects the composition of 1-morphisms up to coherent natural isomorphism.

The coherence isomorphisms consist of natural isomorphisms $$\begin{aligned}
\label{eqnnatisocohuni}
\Phiit_{(S,\xi)}\colon {\text{id}}_{T_\theta(S,\xi)} \Longrightarrow T_\theta({\text{id}}_{S,\xi}) \end{aligned}$$ for all objects $(S,\xi)\in {T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$ and $$\begin{aligned}
\Phiit_{(\Sigmait_a,\varphi_a),(\Sigmait_b,\varphi_b)}\colon T_\theta(\Sigmait_b,\varphi_b) \circ T_\theta(\Sigmait_a,\varphi_a) \Longrightarrow T_\theta((\Sigmait_b,\varphi_b)\circ (\Sigmait_a,\varphi_a))\label{eqnnatisocohasso}\end{aligned}$$ for all composable 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 ${T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$.

Using the enriched co-Yoneda lemma we can write the identity as the coend $$\begin{aligned}
{\text{id}}_{T_\theta(S,\xi)}(?) \cong \int^{\sigma \in {T_\theta(S,\xi)}} {\operatorname{Hom}}_{T_\theta(S,\xi)} (\sigma , ?)* \sigma .\end{aligned}$$ Without lost of generality we can evaluate this at a generator $\sigma_0\in {\operatorname{Fund}}(S_1)$ $$\begin{aligned}
\sigma_0 \cong \int^{\sigma \in {T_\theta(S,\xi)}} {\operatorname{Hom}}_{T_\theta(S,\xi)} ( \sigma , \sigma_0)* \sigma \cong \int^{\sigma \in {{\operatorname{Fund}}(S)}} {\operatorname{Hom}}_{T_\theta(S,\xi)}( \sigma , \sigma_0)* \sigma.\label{cohid1eqn}\end{aligned}$$ On the other hand we have $$\begin{aligned}
T_\theta({\text{id}}_{S,\xi})(\sigma_0) = \int ^{\sigma \in {{\operatorname{Fund}}(S)}}  (S\times [0,1])^{ \xi \times [0,1]}  