We investigated the relationship between tensor-network approaches to fusion categories in two-dimensional topological condensed matter physics and operator algebra theory. It has been known that the 4-tensor describing objects of a fusion category is essentially identical to the notion of a bi-unitary connection in operator algebra theory. This year, we proved that the 3-tensor connecting these 4-tensors and satisfying the zipper condition coincides with the flat field of strings in operator algebra theory, and that it provides the morphisms of the fusion category. While this result is naturally expected, we established it in full generality.