Adding CPTs to the edit.php form

I have two custom post types and several custom taxonomies.

When I’m editing the custom taxonomy, I would like to see a column for each post type that can use it.

To give you a visual, look at (on your site):
https://www.example.com/wp-admin/edit-tags.php?taxonomy=category

See the table on the right where the column headings are:
name description slug posts

Imagine that if I had custom taxonomies ct01 and ct02, then the columns would show:
name description slug posts ct01 ct02

Of course I would want this to work for all post types and their related taxonomies like this:
edit-tags.php?post_type=<any>&taxonomy=<all-relevant>

I see several do_action() handlers in edit-tags.php and edit-tags-form.php. Am I on the right track to devise something that will hook into that?

Thanks all.