This LockDown Browser-Mac error is similar to the LockDown Browser-Windows error discussed in the knowledgebase article, "Update server connection is blocked or not available".
For LockDown Browser-Mac, the "Update Check failed" warning typically indicates security settings or another application is blocking access to the Respondus servers that LockDown Browser accesses at start-up. This can happen when your internet connection fails, or when security settings or another application blocks access to the Respondus servers.
Please verify LockDown Browser can access the following domains at start-up:
These domains may need to be added as "trusted sites" in security settings on the local computer and may need to be added to network firewalls or routers. For Macs, you should check settings in System Preferences->Security & Privacy.
We also recommend doing a complete power down and restart of your Mac.
For Macs, the "a4 error" can be due to permissions issues with the specific user account on the Mac. Some students have found creating another user account with "Allow user to administer this computer", restarting the Mac and logging with the new user account allowed them to start LockDown Browser.