Answer: Coleus is native to a warm climate and so doesn't mind summer heat as long as it receives adequate water and possibly partial shade. It also does well as a house plant in a bright location with average watering and fertility. There are several possible explanations for the symptoms you have described. If it was accustomed to being outdoors, it may be responding to the decreased light levels indoors. (Plants need to adjust slowly to this type of change, generally it is better to move them gradually over a period of weeks into the new type of light.)
It may need repotting, especially if it has grown a lot since you got it. The soil may be exhausted or the roots may be too crowded, or both. It may need some fertilizing, so you could fertilize it regularly with a water soluble fertilizer for foliage house plants according to the label instructions.
You will also need to keep it watered so that the soil is moist but not soggy or sopping wet. Over or underwatering watering can both cause drooping and browning. Note too that it will need less water indoors than it would outdoors.
Finally, take a good close look at the plant for signs of insects. Check the undersides of the leaves and along the stem, too.
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