Backfires in the Morning

I assume in the morning the engine is cold, and it backfires until the engine warms up. This could be due to a faulty engine coolant temperature sensor, so I would try a new one.

In my opinion if it were the fuel injectors it would not matter if the engine were cold or hot.
The engine coolant temperature sensor senses the temperature of the engine, then sends that info to the computer so the computer can adjust fuel injector pulse, etc, to run better for cold or hot engines but if the information is wrong the computer may adjust the fuel injector pulse for what it believes is a warm engine when in fact it is now, it is cold.
I do not have the repair manual that covers a Corolla, but the engine temperature sensor is usually found either screwed into the block, intake manifold or somewhere where it can sense the temperature of the coolant.