Advanced search using a text input to search through only one category

function advanced_search_query( $query ) {
    if( $query->is_search() ) {
        // find the category by slug
        $term = get_category_by_slug( 'case-study' ); 
        // get the category ID
        $id = $term->term_id;
        // set the query argument to search within the category
        $query->set( 'cat', $cat_id );
    }

    // removed "return $query" - $query is passed by reference, so no need to return
}

add_action('pre_get_posts', 'advanced_search_query');

<form method="get" id="advanced-searchform" role="search" action="<?php echo esc_url( home_url( '/' ) ); ?>">
    <input type="hidden" name="search" value="post">
    <input id="search-case-study" class="search-case-study" type="text" value="" placeholder="Search..." name="s" />
    <!-- add a hidden input that passes the desired category slug -->
    <input name="cat_slug" value="case-study" />
    <input type="submit" id="searchsubmit" value="Search" />
</form>

function advanced_search_query( $query ) {
    // check if search AND if "cat_slug" input was present
    if( $query->is_search() && ! empty( $_GET['cat_slug'] ) ) {
        // find the category by the slug passed in the input
        $term = get_category_by_slug( $_GET['cat_slug'] ); 
        // defensive check, in case the category could not be found
        if ( ! empty( $term->term_id ) ) {
            // get the category ID
            $cat_id = $term->term_id;
            // set the query argument to search within the category
            $query->set( 'cat', $cat_id );
        }
    }

    // removed "return $query" - $query is passed by reference, so no need to return
}

add_action('pre_get_posts', 'advanced_search_query');