How to make a post page a category page at the same time?

Question

WordPress

I have a blog page website/blog (in the admin it is not listed as a post page)
On this page, using wp_list_categories, a list of categories is displayed, and below all posts. By clicking on a category, we get to the category page website/category/family. I would like the posts from the selected category to be displayed on the blog page in the same place where all posts are usually displayed. However, going to the category picks up the category.php template, which I have to make exactly the same as blog.php. Is it possible for the specified category to be displayed on website/blog/family, i.e. essentially on the same page?