Unobstructed Immersed Lagrangian Correspondence and Filtered $A_{\infty}$ Functor

In this paper, we ‘construct’ a $2$-functor from the unobstructed immersed Weinstein category to the category of all filtered $A_{\infty}$ categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered $A_{\infty}$ category associated to $(X,\omega)$ is defined by using Lagrangian Floer theory in such generality, see Akaho–Joyce (2010) and Fukaya–Oh–Ohta–Ono (2009). The morphism of unobstructed immersed Weinstein category (from $(X_1,\omega_1)$ to $(X_2,\omega_2)$) is by definition a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho–Joyce (2010) and Fukaya–Oh–Ohta–Ono (2009)). Such a morphism transforms an (immersed) Lagrangian submanifold of $(X_1,\omega_1)$ to one of $(X_2,\omega_2)$. The key new result proved in this paper shows that this geometric transformation preserves unobstructedness of the Lagrangian Floer theory. Thus, this paper generalizes earlier results by Wehrheim–Woodward and Mau's–Wehrheim–Woodward so that it works in complete generality in the compact case. The main idea of the proofs are based on Lekili–Lipyanskiy's Y diagram and a lemma from homological algebra, together with systematic use of Yoneda functor. In other words, the proofs are based on a different idea from those which are studied by Bottmann–Mau's–Wehrheim–Woodward, where strip shrinking and figure $8$ bubble plays the central role.