We extend the skein lasagna theory of Morrison–Walker–Wedrich to 4-manifolds with corners and formulate gluing formulas for 4-manifolds with boundary and, more generally, with corners. As an application, we
develop a categorical framework for a presentation of the skein lasagna module of trisected closed 4-manifolds.
Further, we extend the theory to dimension two by introducing bicategories for closed oriented surfaces and proving a gluing formula for the categories associated with 3-manifolds with boundary.