strips, its proof adapts to any quilted surface.

([@MW Lemma 6.4]) The intersection number $\underline{u}\cdot \underline{R}$ doesn’t change when one perturbs $\underline{u}$ by a homotopy preserving the boundary and seam conditions. Moreover, if $\underline{u}$ is pseudo-holomorphic, $\underline{R}$ almost complex and $\underline{u}$ and $\underline{R}$ intersect transversely, then this number is given by: $$\underline{u} \cdot \underline{R} = 
\sum_{j=0}^k 
\# \{ z_j \in \operatorname{int}(S_j) | u_j(z_j)
\in R_j \} + \frac{1}{2} \# \{ z_j \in \partial S_j | u_j(z_j) \in R_j \} .$$

\[defcat\] We will call ${\textbf{Symp}}$ the following category:

$\bullet$ Its objects are tuples $(M, \omega , \tilde{\omega}  , R, \tilde{J})$ satisfying conditions $(i)$, $(ii)$, $(iii)$, $(iv)$, $(v)$, $(x)$, $(xi)$ and $(xii)$ of [@MW Assumption 2.5], namely:

1.  $(M,\omega)$ is a compact symplectic manifold.

2.  $\tilde{\omega}$ is a closed 2-form on $M$.

3.  The degeneracy locus $R\subset M$ of $\tilde{\omega}$ is a symplectic hypersurface for $\omega$.

4.  $\tilde{\omega}$ is $\frac{1}{4}$-monotone, that is $\left[ \tilde{\omega} \right] = \frac{1}{4} c_1(TM) \in H^2(M; {\mathbb{R}})$.

5.  The restrictions of $\tilde{\omega}$ and $\omega$ to $M\setminus R$ define the same cohomology class in $H^2(M\setminus R; {\mathbb{R}})$.

6.  The minimal Chern number $N_{M\setminus R}$ with respect to $\omega$ is a positive multiple of 4, so that the minimal 