Auto add taxonomy if post has category

Here’s an (untested) example on how to get the saved post’s categories, check if any of them exists in another taxonomy, and terms that exists to the post. Instead of the generic save_post action you can use the post type specific one, save_post_{post_type}. If you want to prevent the check from running every time the post is saved / updated, you can utilize the third parameter the hook provides, which tells you is the action for a new or an existing post.

// hook to post type specific save action
add_action( 'save_post_post', 'studio_code_save_post', 10, 3 );
function studio_code_save_post( $post_ID, $post, $update ) {
    if ( wp_is_post_revision( $post ) ) {
        return;
    }

    // uncomment to prevent code running on post update
    // if ( $update ) {
    //  return;
    // }

    $categories = get_the_terms( $post, 'category' );
    $studio_terms = array();
    if ( $categories && is_array($categories) ) {
        foreach ($categories as $category) {
            // check, if the term exists in another taxonomy
            if ( term_exists( $category->slug, 'studio' ) ) {
                $studio_terms[] = $category->slug;
            }
        }
    }   

    if ( $studio_terms ) {
        // append terms to post
        wp_set_object_terms( $post_ID, $studio_terms, 'studio', true );
    }
}