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.