Hello,
You can update custom attributes the same as updating any other properties of AD objects. Note, however, that you are able to modify Adaxes custom attributes only when operations are performed via Adaxes service. That is, for this purpose, you'll need to use either cmdlets from Adaxes powerShell module or Adaxes ADSI API.
If you are using Adaxes PowerShell Module, you can call the Set-AdmUser cmdlet for this purpose and specify the -AdaxesService parameter when calling it, for example:
Set-AdmUser "CN=John Doe,CN=Users,DC=example,DC=com" -Add @{"adm-CustomAttributeText1"=$HomeEmail} -AdaxesService localhost
If you prefer using the ADSI API, you can do as follows:
$user = $admService.OpenObject("Adaxes://CN=John Doe,CN=Users,DC=example,DC=com", $NULL, $NULL, 0)
$user.Put("adm-CustomAttributeText1", $HomeEmail)
$user.SetInfo()