select box not checked and is not saved

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In the code documentation:

// TEMPLATE ACTION TAG TO BE USED IN THEME
// Usage: do_action('wpse_crowd_cats_form');
// Usage: do_action('wpse_crowd_cats_form', $post_id, $taxonomy );

If the do_action is going to be placed in the loop, then do_action('wpse_crowd_cats_form', get_the_ID()); should work.