Is there a name for arrows of the type a -> a (in Haskell notation) in category theory?

Question

Whats the name of arrows in category theory that have this type:

a -> a

"From a type(?) to another object of the same type"

Or maybe there's no particular name for them?

In other words: Is there a name for the set of all arrows that go from any type a to the same type a? Examples of arrows (functions?) of that set:

\x->x+x   :: Int->Int
\x-> "hello, " ++ x :: String -> String
...

Edit

@leftaroundabout says that I'm using OO definition of object for category theory, which is wrong. Therefore, what I'm really asking is: "In category theory, in a category 𝓒 what is the name for a morphism from some object O of 𝓒 to O itself?"

Answer 1

If I'm correct in interpreting your question as "what do we call morphisms from an object to itself in category theory?", then the word you're looking for is endomorphism.

Answer 2

The word you're looking for, as many others have said, is "endomorphism." But in a more concrete note it's worth mentioning here the Endo type in Data.Monoid:

data Endo a = Endo { appEndo :: a -> a }

instance Monoid (Endo a) where
    mempty = Endo id
    Endo f `mappend` Endo g = Endo (f . g)

This type is sometimes useful. For example, as Brent Yorgey explains, folds are made of monoids:

import Data.Monoid

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f z xs = appEndo (mconcat (map (Endo . f) xs)) z

foldl :: (b -> a -> b) -> b -> [a] -> b
foldl f z xs = appEndo (mconcat (map (Endo . flip f) (reverse xs))) z

So, since monoids are associative, oftentimes folds can be parallelized (with a divide-and-conquer strategy) by first rewriting them in terms of Endo, and then replacing the specific Endo b for that fold with some more concrete type that allows for some of the work to be done at each mappend step.