Add new post only in assigned category

First remove the category metabox from the post edit screen for this particular role, let’s name it onecat-author:

if (current_user_can('onecat-author')) {
  remove_meta_box ('category', 'post', 'normal');   
  }

Next, when the post is saved use the available hook to assign the category:

if (current_user_can('onecat-author')) {
  add_action ('save_post', 'wpse_231846_save_cat', 10, 1 );
  }
function wpse_231846_save_cat ($post_id) {
  wp_set_post_categories ($post_id, array('the-id-of-that-category'));
  }

Note: the code is untested, so some debugging may be necessary.