the gluing boundary, hence they agree.

\[lemmaverticalcomp\] $T_\theta$ preserves the vertical composition of 2-morphisms strictly.

Given two 2-morphisms $(M,\psi) :  (\Sigmait_a,\varphi_a) \Longrightarrow (\Sigmait_b,\varphi_b)$ and $(M',\psi') \colon (\Sigmait_b,\varphi_b) \Longrightarrow (\Sigmait_c,\varphi_c)$ between 1-morphisms $(S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ it suffices to show that for fundamental cycles $\sigma_0$ and $\sigma_1$ of $S_0$ and $S_1$, respectively, the composition of linear maps $$\begin{aligned}
    \Sigmait_a (\sigma_1,\sigma_0) \xrightarrow{T_\theta (M)_{\sigma_1,\sigma_0}} \Sigmait_b (\sigma_1,\sigma_0)   \xrightarrow{T_\theta (M')_{\sigma_1,\sigma_0}} \Sigmait_c (\sigma_1,\sigma_0) \end{aligned}$$ as defined in is equal to $$\begin{aligned}
        \Sigmait_a (\sigma_1,\sigma_0) \xrightarrow{T_\theta (M'\circ M)_{\sigma_1,\sigma_0}}  \Sigmait_c (\sigma_1,\sigma_0) .
        \end{aligned}$$ Picking fundamental cycles $\nu$ and $\nu'$ for $M$ and $M'$ as in Lemma \[lemmagluing2man\] this follows from $$\begin{aligned}
    \langle \psi^* \theta ,\nu\rangle \cdot \langle {\psi'}^* \theta,\nu'\rangle  =\langle   (\psi'\cup \psi)^* \theta,\nu+\nu' \rangle,
    \end{aligned}$$ where $\psi' \cup \psi : M'\circ M {\longrightarrow}T$ is the map obtained by gluing $\psi$ and $\psi'$.

\[lemmahomotopyinvariance\] $T_\theta$ is defined such that it fulfills the axiom of homotopy invariance from Definition \[defhqftext\].

Consider 2-morphisms $(M,\psi), (M,\psi') \colon  (\Sigmait_a,\varphi_a) \Longrightarrow (\Sigmait_b,\varphi_b)$ between 1-morphisms $(S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ with $\psi \stackrel{h}{\simeq} \psi'$ relative $\partial M$. Let $\sigma_0$ and $\sigma_1$ be fundamental cycles of $S_0$ and $S_1$, respectively. Now for $\mu_a \in {\operatorname{Fund}}_{\sigma_0}^{\sigma_1} ({\Sigmait}_a)$ and $\mu_b \in{\operatorname{Fund}}_{\sigma_0}^{\sigma_1} ({\Sigmait}_b)$ we find a fundamental cycle $\nu$ of $M$ adapted to $\mu_a$ and 