Yes, really hot. But you knew that, yet I am going to repeat it again, it's REALLY HOT in the Grand Canyon in June.
My first backpack in the Grand Canyon was in mid June. Bright Angel trail down to the river, one night in the campground, hiked back up. Here are my thoughts on gear based on that trip.
We started our hike while it was still dark, so we could get a few relatively cool hours in. It was fabulous to see the canyon during sunrise, highly recommended. So get each of you a good headlamp so that you can hike comfortably in the dark.
I didn't bring a shelter at all that first trip and slept relatively well for a few hours at Bright Angel Campground. On a subsequent trip to the Grand Canyon I realized that there are lots of rodents scurrying around the campsites during the night. If that bothers you maybe a fully enclosed tent is in order.
I brought a down sleeping bag and a thermarest. Was thankful for the thermarest but didn't end up using the bag, the night was just to warm. On the other hand not sure I'd want to leave the sleeping bag behind. A conundrum.
I brought a stove and lots of food. Next time I'm bringing less heavy dinners and many more salty snacks and electrolite mixes.
A couple of my hiking mates got bad blisters. Check you feet often, bring several pairs of socks so you can switch out damp ones, and a good blister kit.
A gallon of water capacity per person should be fine considering your hiking on the corridor trails. Make sure each of you have a small bottle that you use for Gatorade type drinks.