Once the Peppers and Eggplants get to their second set of real leaves, I move them to larger pots. This will build bigger and more robust root growth which will get them off to a good start once they are put out into the garden. Once the young plants are transferred to larger pots, they are then put back under the lights in a warm room. They will be left there for about a week to get them acclimated to their new environment. I will then move them into an outdoor cold frame assuming the weather is not too cold at night. You should also provide some shade on those very sunny days and venting on the green house so that they don’t get too much sun or heat. I have a max/min thermometer in the cold frame. During our sunny days which hit the high 60’s we could get in the 90’s in the cold frame. this is much too hot for the young seedlings.
Remember, Peppers and Eggplants are much more sensitive to low temperatures than tomatoes. Eggplants the most so. If you plant out your peppers and eggplants when the soil is too cool, it can greatly slow their growth and sometimes for eggplants I’ve had them not ever recover. It is much better to delay your planting out until the soil is warm and hold off the plants in cold frames. About 60 degrees is a good number. They will catch up, don’t get too anxious.