  We construct the defining data of two-dimensional topological field theories
(TFTs) enriched by non-invertible symmetries/topological defect lines. Simple
formulae for the three-point functions and the lasso two-point functions are
derived, and crossing symmetry is proven. The key ingredients are
open-to-closed maps and a boundary crossing relation, by which we show that a
diagonal basis exists in the defect Hilbert spaces. We then introduce regular
TFTs, provide their explicit constructions for the Fibonacci, Ising and
Haagerup $\mathcal{H}_3$ fusion categories, and match our formulae with
previous bootstrap results. We end by explaining how non-regular TFTs are
obtained from regular TFTs via generalized gauging.
