they correspond to points $(x_1, \cdots , x_k)$ of $\widetilde{L}_{(0)} \cap \widetilde{L}_{(1)},$ with $x_i = p_i(0)$.

Let $\widetilde{\mathcal{M}}(\underline{x},\underline{y})$ be the set of maps $u_i\colon {\mathbb{R}}\times [0,1]\to M_i$ such that, with $s\in{\mathbb{R}}$ and $t\in [0,1]$:

$$\begin{cases} 0 = \partial_s u_i + J_t ( \partial_t u_i - X_{H_i}) \\
\mathrm{lim}_{s\rightarrow - \infty}{u_i(s,t)} = \varphi_i^t (y_i)\\
\mathrm{lim}_{s\rightarrow + \infty}{u_i(s,t)} = \varphi_i^t (x_i)\\
(u_i(s,1), u_{i+1}(s,0) )\in L_{i(i+1)} \\
\underline{u} \cdot \underline{R} = 0 \\
I(\underline{u}) = 1.
\end{cases}$$ The space of quilted Floer trajectories is then the quotient $$\mathcal{M}(\underline{x},\underline{y})  = \widetilde{\mathcal{M}}(\underline{x},\underline{y}) /{\mathbb{R}}$$ by the $s$-reparametrization. For generic choices of Hamiltonians $\underline{H}$ and almost complex structures $\underline{J}$, it is a finite set, see [@WWqfc Prop. 5.2.1].

The Floer complex is then defined as $$CF(\underline{L},\underline{H}, \underline{J}) = \bigoplus_{\underline{x}\in {\mathcal{I}}_{\underline{H}}(\underline{L})} {\mathbb{Z}}\underline{x},$$ and endowed with the differential defined by $$\partial \underline{x} = \sum_{\underline{y}} \# \mathcal{M}(\underline{x},\underline{y}) \underline{y},$$ where $\# \mathcal{M}(\underline{x},\underline{y}) = \sum_{u\in \mathcal{M}(\underline{x},\underline{y}) }{o(u)}$, with $o(\underline{u}) = \pm 1$ the orientation of the point $\underline{u}$ constructed in [@WWorient] with the unique relative spin structure on $\underline{L}$. Recall the following result from Manolescu and Woodward that makes Floer homology well-defined for elements of $Hom_{Symp}(pt,pt)$:

([@MW Theorem 6.5]) Let $\underline{L}$ and $\underline{H}$ be as above. There exists a comeagre subset $$\mathcal{J}_t^{reg}(M_i, 