By the bottom left panel of the remix interface you will see the plugin tool,
click it and search for "CONTRACT VERIFICATION" and activate it.

once activated you will have a new tool on the let hand panel with a check mark symbol.

Click the tool and input the details (Chain,Contract address and contract name) of the smart contract you deployed.
check all the boxes under the "verify on:" statement.

Hit verify and wait for a confirmation.

that's it.