It is hard to generalize, but I can point out a few things belayed on the image. I have only hunted CO.
I hunted sheep in canyons where the rim elevations were 5-6k ft a couple of years ago. This was my first experience hunting this type of area with the canyons. I was looking for sheep but we saw plenty of deer at the spot in southern CO.
The area to the bottom of the photo on the rim (east rim) where there are juniper or similar may be a good area to glass if you can get above it. Deer will hang in the shrub areas at times. On my hunt we did see a lot of deer in the canyons also near the water. This was about the only fresh water available. We would see them bedded alongside the banks of the stream under rock outcroppings or grazing along the stream also.
If you do hunt the canyons, sit and glass. Glass for as long as you like, but we glassed for sheep for hours from the same spot once we set up. We saw deer in the bottoms frequently tucked in shady spots. They were actually easier to spot than sheep because they would travel the bottom along the water where as the sheep would stick right below the cliffs and at times were right under us. The elk and deer hunters were not in the canyons and I imagine this is part of why some were down there. We saw elk down there too.
If you do hunt the canyons, mark routes to the bottom on your GPS as you see them. You may need to hike a couple miles along the rim before finding a viable path down.
I am sure others have some advice, but hopefully you have some luck. If I hunted the sheep unit for deer I would personally stick to the sage and juniper on top. Packing out of a canyon flat out sucks. Would I do it again, probably.
. But it would have to be a nice deer. While walking between glassing points we jumped a lot of deer on the rim of the canyons that were grazing in the sage/juniper areas, but the biggest bucks we saw were in the canyon or right below steep hillsides they could climb/escape in a hurry if something bumped them.