2.  We will sometimes identify $\underline{S}$ and the surface obtained by gluing all the patches together along the seams.

Floer trajectories involved in defining the differential can then be seen as quilted strips as in figure \[bande\].

Weinstein suggested in [@weinsteincat] that Lagrangian correspondences should be seen as morphisms of a category, for which the composition should be defined in the following way:

Let $M_0$, $M_1$, $M_2$ be three symplectic manifolds, and $L_{01} \subset M_0\times M_1$, $L_{12} \subset M_1\times M_2$ be Lagrangian correspondences. The *geometric composition* of $L_{01}$ with $L_{12}$ is the subset: $$L_{01} \circ L_{12} = \pi_{02}(L_{01}\times M_2 \cap M_0\times L_{12}),$$ where $\pi_{02}$ denotes the projection $$\pi_{02}\colon M_0\times M_1\times M_2 \rightarrow M_0\times  M_2 .$$

This composition generalises the composition of symplectomorphisms, and the diagonal $\Delta_{M} = \lbrace (x,x)~|~ x\in M \rbrace$ plays the role of the identity.

Unfortunately, the resulting correspondence may not be smooth, therefore this composition is not always a Lagrangian correspondence. To overcome this difficulty, Wehrheim and Woodward only allow this operation whenever the following criterion is satisfied:

A geometric composition $L_{01} \circ L_{12}$ is called *embedded* when:

-   $L_{01} \times M_2$ and $M_0 \times L_{12}$ intersect transversally,

-   $\pi_{02}$ induces an embedding of $L_{01}\times M_2 