Populate the WordPress post tags with the values from the WordPress post categories on the save_post action hook.

dynamic-post-tags.php

<?php

/**
 * Dynamically set the post tags to mirror the post categories.
 *
 * @action save_post
 *
 * @param  integer $post_id The post ID.
 */
function dynamic_post_tags( $post_id ) {

	// Return if this is not a 'post' or it is a revision
	if ( 'post' !== get_post_type( $post_id ) || wp_is_post_revision( $post_id ) ) {

		return;

	}

	// Retrieve the categories for this post
	$categories = wp_get_post_categories( $post_id );

	// Return if no categories are assigned
	if ( empty( $categories ) ) {

		return;

	}

	// Replace category IDs with the proper category names
	$categories = array_map( function( $category ) {

		$tax_object = get_term( $category, 'category' );

		return $tax_object->name;

	}, $categories );

	/**
	 * Assign the categories to the post tags.
	 *
	 * Note: Argument 4 dictates if tags will be appended, or replace existing tags
	 */
	wp_set_post_terms( $post_id, $categories, 'post_tag', false );

}
add_action( 'save_post', 'dynamic_post_tags', 20 );