Yes, you only need to keep Form FTB 8453-C with your tax records.
According to C, Corporation Responsibilities in the 2023 Instructions for Form FTB 8453-C:
After the corporation’s return is e-filed, the corporation should retain the following documents for the California statute of limitations period:
• Form FTB 8453-C (signed original or copy of the form).
• A paper copy of Form 100, California Corporation Franchise or Income Tax Return; Form 100W, California Corporation Franchise or Income Tax Return - Water’s-Edge Filers; or Form 100S, California S Corporation Franchise or Income Tax Return, and all supporting schedules.
• A paper copy of the corporation’s federal tax return.
The general California statute of limitations is the later of four years from the due date of the return or four years from the date the return is filed.
Exception: An extended statute of limitations period may apply if the corporation’s California or federal tax returns are related or subject to a California or federal audit.