The sups are on install mode so my question is the following: Once I put the new supervisor, does it upgrade automatically like the sups of 6500?

Yes, because you are already running the device in Install Mode. Beside that requirement, you also want to make sure the auto-upgrade feature is enabled, when you do SW# show run all | inc software auto-upgrade you want to see the line "software auto-upgrade enable" listed.

which will be the correct procedure?

The steps for that procedure are on this link below.

Replace a Supervisor Module or Stack Member of Catalyst 9000 in HA

Replace a Redundant Supervisor of C9400 Standalone Chassis

https://www.cisco.com/c/en/us/support/docs/interfaces-modules/catalyst-9600-series-supervisor-engine-1/216193-replace-a-supervisor-module-or-stack-mem.html#anc15