wp_ajax_inline_save_tax

function wp_ajax_inline_save_tax() {
	check_ajax_referer( 'taxinlineeditnonce', '_inline_edit' );

	$taxonomy = sanitize_key( $_POST['taxonomy'] );
	$tax      = get_taxonomy( $taxonomy );

	if ( ! $tax ) {
		wp_die( 0 );
	}

	if ( ! isset( $_POST['tax_ID'] ) || ! (int) $_POST['tax_ID'] ) {
		wp_die( -1 );
	}

	$id = (int) $_POST['tax_ID'];

	if ( ! current_user_can( 'edit_term', $id ) ) {
		wp_die( -1 );
	}

	$wp_list_table = _get_list_table( 'WP_Terms_List_Table', array( 'screen' => 'edit-' . $taxonomy ) );

	$tag                  = get_term( $id, $taxonomy );
	$_POST['description'] = $tag->description;

	$updated = wp_update_term( $id, $taxonomy, $_POST );

	if ( $updated && ! is_wp_error( $updated ) ) {
		$tag = get_term( $updated['term_id'], $taxonomy );
		if ( ! $tag || is_wp_error( $tag ) ) {
			if ( is_wp_error( $tag ) && $tag->get_error_message() ) {
				wp_die( $tag->get_error_message() );
			}
			wp_die( __( 'Item not updated.' ) );
		}
	} else {
		if ( is_wp_error( $updated ) && $updated->get_error_message() ) {
			wp_die( $updated->get_error_message() );
		}
		wp_die( __( 'Item not updated.' ) );
	}

	$level  = 0;
	$parent = $tag->parent;

	while ( $parent > 0 ) {
		$parent_tag = get_term( $parent, $taxonomy );
		$parent     = $parent_tag->parent;
		$level++;
	}

	$wp_list_table->single_row( $tag, $level );
	wp_die();
}