In Haskell and PureScript, `fmap` is a method in the `Functor` class, which is defined by

   class Functor f where

     fmap :: (a -> b) -> f a -> f b

for which instances should satisfy `fmap id = id` (and you get `fmap (g . f) = fmap g . fmap f` for free). If you can find a type for which you can implement `fmap` and satisfy the laws, then you have an instance of `Functor`.

That's the precise and short definition. It assumes the following knowledge:

0. How partial application works and the . works.

1. How Haskell's type-classes work (`Functor`).

2. How Haskell's methods work (`fmap`).

3. How Haskell's higher-kinded types work (the `f`).

So good luck explaining `fmap` in one sentence unless you already know Haskell and understand its type system to a confident degree, which depending on your background can take weeks to months.

Agree. Also, order can improve. You talk about pure functions on the first sentence. Later you explain what a pure function is. Explaining pure functions before monads can help. Same case for functors.

Or morphism.