There are functions for retrieving a term/category link (i.e. URL to the term archive page) and in the case of the default category taxonomy, you can use get_category_link():
<a href="https://wordpress.stackexchange.com/questions/361461/<?php echo esc_url( get_category_link( $category[0] ) ); ?>"><?php echo $category[0]->name; ?></a>
For custom taxonomies, you’d use get_term_link().
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