$L$ is compatible with an hypersurface $R$ if and only if $L$ and $R$ are disjoint.

Let $\underline{S}$ be a quilted surface, $\underline{u}\colon \underline{S}\rightarrow (\underline{M},\underline{L})$ a quilt, and $\underline{R}\subset \underline{M}$ a family of hypersurfaces such that each Lagrangian correspondence associated to a seam is compatible with the corresponding hypersurfaces. Recall that $\underline{u}$ and $\underline{R}$ have a well-defined intersection number $\underline{u}. \underline{R}$:

Let $\underline{U}\subset \underline{S}$ be an open neighborhood of $\underline{u}^{-1}(\underline{R})$ such that the image of each patch $S_i$ is contained in the tubular neighborhoods $\tau_i$ of $R_i$. Each map $u_i$ can be seen as a section of the complex line bundle $u_i^* N_{R_i}$. All these bundles can be glued together into a bundle over $\underline{U}$ by using the isomorphisms $\varphi$, and the sections $u_i$ glue together to a global section of this bundle, which is nonzero over the boundary $\partial \underline{U}$. The bundle is then trivial over this boundary and extends to a bundle over $\underline{S}$, and the sections extend to global sections, which are non-zero outside $\underline{U}$. The intersection number $\underline{u}\cdot \underline{R}$ is then defined as the Euler number of this bundle.

The following lemma is proven in [@MW Lemma 6.4] when the quilted surface consists of several parallel 