Starter does not engage. I replaced the battery

I remember on several Toyota Corolllas, the starter solenoid would throw the contact plunger back and make contact from Battery to Starter before the Pinion gear on the starter shaft was fully engaged with the Flywheel Gear. Check for wear on the actuating fork pivot point (Bearing), that throws the gear to the back engaging the ring gear on the flywheel, and makes the contact plate short the terminals to provide current to the starter motor

Check: the ground wiring, power to the starting relay, the solenoid, the starter???

You have bought a faulty Starter, Replace it

check hot wire to starter from battery also in line fuse

Well since you replaced the starter, I think I would load test the battery cold and then again hot. It could be you have a dead cell in the battery and it is not producing enough voltage to work the starter solenoid.
Or it could be a bad connection between the battery and starter, or battery and ground.

Check power to starter motor solenoid, solenoid operation and starter motor voltage during cranking. If the starter motor is receiving power from the solenoid, but isn't cranking, then replace the starter motor.

Also check the condition of battery terminals, wiring and connections. Also check the earth wire from the body to the engine.

this could be a few problems that cause this,

1. your battery could be weak and not have enough juice to engage the starter solenoid

2. the bendix could be messed up

3. this is probaly the most common problem, the starter solenoid went out

the best course of action would be to have your battery load tested, and if its fine, replace the starter

If the starter motor remained engaged when you released the key, the underlying cause is likely a bad starter solenoid (first item the positive battery cable connects to after the battery) Remaining engaged, can damage the starter itself. Replace the solenoid, check the starter and you should be finished. (Or also replace the starter if it was damaged)
If the starter was stuck and not engaging, the bendix drive inside the starter has failed...In that case it's best to change the starter as even though it's replaceable, generally by the time it needs replacement the starter is worn in other areas and would need complete overhaul.

Replace the starter,that`s the only way to fix the problem.