for defining [**Symp**]{} (c.f. Definition \[defcat\]). We first give here some definitions required in order to do so.

As the symplectic manifolds we will consider will be equipped with symplectic hypersurfaces, the Lagrangian correspondences shall satisfy the following compatibility condition ([@MW Def. 6.2]):

\[correspcompat\] Let $M_0$, $M_1$ be two symplectic manifolds, and $R_{0}\subset M_0$, $R_{1} \subset M_1 $ two symplectic hypersurfaces. A Lagrangian correspondence $L_{01} \subset M_0\times M_1$ is *compatible with the pair $(R_0,R_1)$* if $L_{01}$ intersects $R_0 \times M_1$ and $M_0 \times R_1$ transversally, and these two intersections coincide (and are equal to $(R_0 \times R_1 ) \cap L_{01}$).

In these conditions, one can choose tubular neighborhoods $\tau_0$ and $\tau_1$ of $R_0$ and $R_1$ respectively: $$\tau_0\colon N_{R_0} \rightarrow M_0,\   \tau_1\colon N_{R_1} \rightarrow M_1,$$ such that the preimage $ (\tau_0 \times \tau_1)^{-1}(L_{01}) \subset N_{R_0} \times N_{R_1} $ is the graph of a bundle isomorphism $$\varphi \colon (\widetilde{N}_{R_0})|_{(R_0 \times R_1 ) \cap L_{01}} \rightarrow (\widetilde{N}_{R_1})|_{(R_0 \times R_1 ) \cap L_{01}},$$ where $\widetilde{N}_{R_0}$ is the normal bundle of $R_0 \times M_1 \subset M_0 \times M_1$, and $\widetilde{N}_{R_1}$ is the normal bundle of $M_0 \times R_1 \subset M_0 \times M_1$.

Notice that if one of the symplectic manifolds is a point, a Lagrangian 