  Min-max theory for the Allen-Cahn equation was developed by Guaraco and
Gaspar-Guaraco. They showed that the Allen-Cahn widths are greater than or
equal to the Almgren-Pitts widths. In this article we will prove that the
reverse inequalities also hold i.e. the Allen-Cahn widths are less than or
equal to the Almgren-Pitts widths. Hence, the Almgren-Pitts widths and the
Allen-Cahn widths coincide. We will also show that all the closed minimal
hypersurfaces (with optimal regularity) which are obtained from the Allen-Cahn
min-max theory are also produced by the Almgren-Pitts min-max theory. As a
consequence, we will point out that the index upper bound in the Almgren-Pitts
setting, proved by Marques-Neves and Li, can also be obtained from the index
upper bound in the Allen-Cahn setting, proved by Gaspar and Hiesmayr.
