Custom Roll/Custom Post Type – Can’t Select Categories

I had the same problem, a custom role couldn’t assign categories to my CPT.
When doing register_taxonomy(), I added these capabilities:

            'capabilities' => array (
                'manage_terms' => 'manage_options', //by default only admin
                'edit_terms' => 'manage_options',
                'delete_terms' => 'manage_options',
                'assign_terms' => 'edit_cpt-type'  // can edit cpt-type i.e. custom role
        ),

Hope this helps to people and cyborgs having this problem.

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