R_{i+1} )$-compatible, spin outside $R_{i-1}\times R_{i+1}$, satisfies the above hypothesis concerning pseudo-holomorphic discs, and also the following one: every quilted pseudo-holomorphic cylinder as in figure \[cylinder\] of zero area and with seam conditions in $L_{(i-1)i}$, $L_{i(i+1)}$ and $L_{(i-1)i}\circ  L_{i(i+1)}$ has an intersection number with $(R_{i-1} , R_i, R_{i+1})$ smaller than $-2$.

\[monotonicity\] Under these hypotheses, the generalized Lagrangian correspondences are automatically monotone: if $\underline{x},\underline{y} \in  {\mathcal{I}}(\underline{L})$ are generalized intersection points, and $\underline{u}$ denotes a quilted strip with seam conditions in $\underline{L}$ and with limits $\underline{x},\underline{y}$, then the symplectic area of $\underline{u}$ is $A(\underline{u}) = \frac{1}{8} I(\underline{u}) + c(\underline{x},\underline{y})$, with $c(\underline{x},\underline{y})$ a number only depending on the points $\underline{x}$ and $\underline{y}$. This follows from monotonicity of the symplectic manifolds and from simply connectedness of the correspondences, see [@MW Lemma 2.8].

### Definition of quilted Floer homology  {#defhomolgiequilted}

Let $\underline{L} = (L_{i(i+1)})_{i = 0\cdots k} $ be a morphism of ${\textbf{Symp}}$ of length $k+1$ from $pt$ to $pt$, with intermediate objects $\underline{M} = (M_i, \omega_i , \tilde{\omega}_i, R_i , \tilde{J}_i)_{i = 0\cdots k+1}$, and $M_0 = M_{k+1} = pt$:

$$\underline{L} = \left(  \xymatrix{   pt\ar[r]^{L_{01}} &  M_1 \ar[r]^{L_{12}} &  M_2 \ar[r]^{L_{23}} & \cdots \ar[r]^{L_{k(k+1)}} & pt } \right).$$

Let $\mathcal{J}(M_i, \mathrm{int}\left\lbrace \omega_i = 