Hey there claudefromreconvilier,
Thanks for reaching out to the Apple Support Communities. To get started, from the screenshot of the alert the recipient is seeing, the best option would be to try and accept the calendar invite while logged into their Apple ID from a computer at this link here: iCloud - Apple
For more help with accepting a shared calendar invitation on iCloud.com, provide the recipient with these steps below: Accept a shared calendar invitation on iCloud.com - Apple Support
1. In Calendar on iCloud.com, click below the calendar. If you have one or more pending invitations, the button shows a number instead of an arrow.
2. In the calendar notification, click Join Calendar.
We hope that's able to help.
Have a great day!