Search by tag name and category

use this function put in function.php file for cat=22 filter change your cat id for you mind

function my_search_form( $form ) { 

    $form = '<form role="search" method="get" id="searchform" action="' . home_url( "https://wordpress.stackexchange.com/" ) . '" >
    <div><label class="screen-reader-text" for="s">' . __('Search for:') . '</label>
    <input type="text" value="' . get_search_query() . '" name="s" id="s" />
   <input type="text" value="22" name="cat" id="scat" />
    <input type="submit" id="searchsubmit" value="'. esc_attr__('Search') .'" />
    </div>
    </form>';

    return $form;
}

add_filter( 'get_search_form', 'my_search_form' );