that the geometric composition $L_{i-1,i} \circ  L_{i, i + 1}$ is embedded, simply connected, $(R_{i-1} , R_{i+1})$-compatible, and such that the intersection number of every pseudo-holomorphic quilted cylinder with $(R_{i-1} , R_i,  R_{i+1})$ is smaller than $-2$, then $HF (\underline{L})$ is canonically isomorphic to $HF (\cdots L_{i-1,i} \circ L_{i, i + 1} \cdots )$.

Cobordisms with vertical boundaries and connected Cerf theory
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\[sec:connectedcerf\]

We start by defining the cobordism category that will be the source of the Floer field theory functor.

\[chemincob\] We will call *category of cobordisms with vertical boundaries, with degree one ${{\mathbb{Z}}_{2}}$-homology class*, and will denote it ${\textbf{Cob}}$, the category whose:

-   objects are pairs $(\Sigma,p)$, where $\Sigma$ is a compact connected oriented surface, with connected boundary, and $p\colon {\mathbb{R}}/{\mathbb{Z}}\rightarrow \partial\Sigma$ is a diffeomorphism (parametrization).

-   morphisms from $(\Sigma_0,p_0)$ to $(\Sigma_1,p_1)$ are diffeomorphism classes of 5-tuples $(W, \pi_{\Sigma_0},  \pi_{\Sigma_1}, p,  c)$, where $W$ is a compact oriented 3-manifold with boundary, $\pi_{\Sigma_0}$, $\pi_{\Sigma_1}$ and $p$ are embeddings of $\Sigma_0$, $\Sigma_1$ and ${\mathbb{R}}/{\mathbb{Z}}\times [0,1]$ into $\partial W$, the first reversing the orientation, the two others preserving it, and such that:

    -   $\partial W = \pi_{\Sigma_0}(\Sigma_0)\cup  \pi_{\Sigma_1}(\Sigma_1)\cup  p({\mathbb{R}}/{\mathbb{Z}}\times [0,1])$,

    -   $\pi_{\Sigma_0}(\Sigma_0)$ and $\pi_{\Sigma_1}(\Sigma_1)$ are disjoint,

    -   for $i=0,1$, $p(s,i) 