You are mostly right. Haskell Functors, when thought of as mathematical functors do go from Hask to Hask and thus are endofunctors. But you also say that "Hask

You are mostly right. Haskell Functors, when thought of as mathematical functors do go from Hask to Hask and thus are endofunctors. But you also say that "Hask is the only category". That needs some qualification, since the type class Control.Category is also in common use and models mathematical categories. Notice that the standard Functor class doesn't use Control.Category to specify the domain and target, so what you said about Haskell Functors modelling functors Hask -> Hask is correct.