Flight Instructor reported NMAC on climbout; with traffic turning crosswind early; resulted in the need for evasive action.

Narrative

While on a VFR departure off of Runway 22R at FFZ and In the up-wind and turning for Left X-Wind; another aircraft (Aircraft Y) was on the [simulated departure] behind me and turned their XW (Crosswind) inside of my turn and was flying on a converging course and climbing underneath me. The vertical separation was less than 100 feet; the horizontal separation was less than 100 feet and from my estimation the actual distance was approx 100-150 feet straight-line; I could clearly read the tail number [of] Aircraft Y. The approximate time of the incident was XA01Z.As soon as I aware of the situation I took controls from my SP (Student Pilot) and made evasive corrections to gain separation and avoid a mid-air collision. I then contacted Tower and asked if the aircraft had clearance for an early left XW; to which they responded no and that they were supposed to be on a [simulated departure]. whose turn happens at three dome church a specified Visual Waypoint and not at a specified altitude like in the ODP. Shortly after the pilot of Aircraft Y called and said he turned at the altitude; to which Tower responded that for the [simulated departure] the turn happens that the visual waypoint not a specified altitude. I have no doubt that if I had not acted when I did that we would have had a mid-air collision.