Steps to your SuisseID
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This is a characteristic of Safari. In the first connection attempt with your SuisseID, Safari offers you a choice of certificates which are on your SuisseID. If the user does not choose the authentication certificate, the connection fails.Unfortunately this choice is also stored in Safari for future connections. There is also the issue that the wrong certificate is chosen by default, i.e. when the user simply clicks on OK.
This procedure is described in detail on the Apple Website.