Given the heat we are experiencing, I would hope no lower than 3 inches when cutting your lawn. In fact, 3 1/2 is preferable.
It amazes me when I see turf scalped to a putting green in extreme conditions like our current summer heat and humidity.
No matter how much you water, you are just asking for problems in the way of insects, weeds, crabgrass and burning from the sun.
For some, leaving grass longer for drought tolerance and overall health looks unsightly and unkempt, but it doesn't have to be that way.
A lawn cut at 3 1/2 inches, to me, looks much better and healthier than one shaved to barely an inch above the soil.
I much prefer a healthy green (long) lawn than one going dormant and the colour of straw. Generally the higher the grass the less water you will need on a weekly basis.
Ultimately it is a choice of personal preference, but keep in mind; if you aren't going to practice proper cutting techniques under extreme heat conditions, it really doesn't matter what your lawn care provider does. The results will be the same...absolutely nothing.