I would discourage talk of "the Hask Category" because it subconsciously conditions you against looking for other categorical structure in Haskell programming.
Note, I dislike the use of "Hask" as the name of the "category of Haskell types and functions": I fear that labelling one category as the Haskell category has the unfortunate side-effect of blinding us to the wealth of other categorical structure in Haskell programming. It's a trap.
I wish people wouldn't call it "Hask", though: it threatens to limit the imagination.
What other categories can we see in Haskell?
In one of his answers, he touches upon some of these ideas, but I wonder if someone could expand upon it; and I wonder if there are even more examples.
[...] there's a ton of categorical structure lurking everywhere, there's certainly a ton of categorical structure available (possibly but not necessarily) at higher kinds. I'm particularly fond of functors between indexed families of sets.
Gabriel Gonzalez has blogged about this. Here's one such post: http://www.haskellforall.com/2012/08/the-category-design-pattern.html
In it, he calls Hask "the function category", and also discusses "the Kleisli category" and "the pipes category." These are all examples of instances of the Category typeclass in Haskell. The Category typeclass in Haskell is a subset of the categories you can find in Haskell.