This is just restating the joke.

I'm really high and I really don't mean this to come across as critical but what was the thought process in adding the s to bateman? If you skip the s the joke still makes sense and you get to keep the last name intact.

The downy is more sparrow-sized, while I think the hairy is more between the size of a cardinal and blue jay, but in this picture you'd use the bill. The hairy's bill is much larger in proportion to its head.

Ah, Hairy Woodpecker it is then