question about exhaust fan

i closet grow and i have an exhaust fan always on but the temp in the closet stays around 84 degrees or so, my question is what if i have a fan pushing air from my colder room into the warmer closet? since hot air rises the cooler air should be around the plants and the warmer air should escape towards the top of the door frame above the fan correct? the air would still be circulating in and out of the closet, has anyone ever tried this instead of just an exhaust fan?

You obviously have a grow box, but what kind of fan are you using and what size is the box, what lights ect.

As for the idea of blowing cold air it would work but not very well. Unless you have a fan pushing air into your grow box that is cold you will probably run into problems heat still. I just leave my closet door open during the day. During the night the room cools down naturally and I close the door.

8 42 watt cfls and 4 54 watt cfls, i wanna switch to hps/mh but if my closet already gets in the mid 80s i would assume it would get hotter thats why i have yet to switch and idk the size of my closet exactly but i only have 2 plants in there along with an oscillating fan inside with a box fan at the door way acting like an exhaust fan, my ac is on all day since it gets pretty hot where i live, i always keep my door open all the time but now that im in flowering it shuts when the lights go out and im not using a grow box, its a spare closet with nothing in it except my babys....so your saying just stick with an exhaust fan instead of the other way around?