Replaced battery on 2002 camry battery light still stays on ??

Question edited for clarity.

It might be a BCM issue

https://parts.toyota.com/p/Toyota__Camry/Body-Control-Module/63378233/8922006091.html

https://moduleexperts.com/5-common-symptoms-failing-body-control-module-bcm/

or fuses

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Toyota Camry (2002 - 2003) - fuse box diagram - Auto ...

https://www.autogenius.info > toyota-camry-2002-2003...
Jul 31, 2016 - Toyota Camry (2002 - 2003) - fuse box diagram · Engine compartment · Engine compartment (with vehicle skid control system) · Instrument panel.

Fuse Box Diagram Toyota Camry (XV30; 2002-2006)

https://fuse-box.info > toyota > toyota-camry-xv30-200...
The fuse box is located in the instrument panel, behind the cover to the left of the steering wheel. The location of the fuses in the passenger compartment: ...

Also check for codes with a Scan tool.

check the output of your alternator it may be low ,and check the belt as it may be slipping .the fact that when you put on your AC shows the trouble is a slipping belt, do that first, then check the output of the alternator.

Go to Autozone and have them test your battery and alternator,make sure cable from positive battery cable to starter is tight and negative cable is tight at other end and battery terminals are clean and tight.could be a dead cell or alternator is over charging..

if the indecators stay flashing permenant after unlocking then there is a switch that is closed circuit perminantly on or there is a loose earth more than likely a lose earth after changing the battery

you need to reset the alarm/immobiliser each time the power is disconnected.....

recharge battery, it must be dead

if light stays on it means the problem is an active code meaning what ever the cause it is broken, you need to drive it over to auto zone or advance and have them pull the codes for you they do it for free.

Was it idling before you changed the battery? If you were to clean out the throat of the aluminum inlet manifold just behind the throttle body with a parts cleaner product that is safe for plastics (sensors) the car will idle.
The engine features a intake valve cover vent that goes back into the front of the throttle body. After time, the vent tube deposits a lot of oil and gunk in the throttle body assembly and just behind it into the intake manifold.

You need to readjust the switch on your trunk door, or replace. To check it locate it, tape it down with duct tape and see if the light goes off. If not it's bad. If it does it means it isn't sticking out far enough to be turned off when the trunk is closed. The adjusment should be fairly obvious. If not you can simply unplug it which has the same result as turning it off.

Replace the door ajar switch that is defective, short answer. But the issue here is any of the doors can cause the problem, since in order to replace this tiny switch the door panel must be removed to access the door latch the smart thing to do is have the dealer scan the BCM (body control module) and see which switch is causing the problem. Cost will be about $300.00 depending on where you live