How to add sort order to incremented and paginated category loop

Add pagination parameters to WP_Query,
Check Here.

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Please check your query:
$sortorder= new WP_Query( 'orderby=date&sort=asc' );

There is no parameter of posts_per_page and paged, so it will get the same ( 1st ) page result only each time and display the same data.

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If you want to manage just some parameters only in the main query, do not use custom query use pre_get_posts hook.

Please check:

<?php
function custom_posts_per_page($query) {
    if( !is_admin() && $query->is_category()) {
      $qobj = get_queried_object();
      if( isset($qobj->taxonomy) && 'category' == $qobj->taxonomy ) {
        $query->set( 'orderby', 'date' );
        $query->set( 'order', 'asc' );
      }
    }
}
add_filter( 'pre_get_posts', 'custom_posts_per_page' );