type of high order functions

if I specify the (I think) correct type for a high order function the OCaml compiler rejects the second usage of that function.

The code

let foo ():string  =
    let f: ('a -> string) -> 'a -> string = fun g v -> g v
    in let h = string_of_int
    in let i = string_of_float
    in let x = f h 23
    in let y = f i 23.0
    in x ^ y

leads to the following error message

File "test.ml", line 6, characters 14-15:
Error: This expression has type float -> string
       but an expression was expected of type int -> string

So the first usage of f seems to fix the type of its first parameter to int -> string. I could understand that. But what I don't get is that omitting the type restriction on f fixes the problem.

let foo ():string  =
    let f g v = g v
    in let h = string_of_int
    in let i = string_of_float
    in let x = f h 23
    in let y = f i 23.0
    in x ^ y

And moving f to global scope fixes the problem, too:

let f: ('a -> string) -> 'a -> string = fun g v -> g v

let foo ():string  =
  let h = string_of_int
  in let i = string_of_float
  in let x = f h 23
  in let y = f i 23.0
  in x ^ y

Why is it that the first example does not compile while the later ones do?