The unified mathematical theory of gapped and gapless edges of two-dimensional (2d) topological orders was developed by two of the authors. According to this theory, the critical point of a purely edge topological phase transition of a 2d topological order can be mathematically characterized by an enriched fusion category. In this work, we provide a physical proof of this fact in a concrete example: the 2d Z2 topological order. In particular, we construct an enriched fusion category, which describes a gappable nonchiral gapless edge of the 2d Z2 topological order. Then, we use an explicit lattice model construction to realize a topological phase transition between the two well-known gapped edges of the 2d Z2 topological order, and show that all the ingredients of the above enriched fusion category can be realized explicitly in this lattice model. © 2020 American Physical Society.