Is an identity functor in Category of sets a function application?

Question

https://ncatlab.org/nlab/show/identity+functor

The identity functor on a category C is the functor idC:C→C that maps each object and morphism of C to itself. The identity functors are the identities for composition of functors in Cat.

https://ncatlab.org/nlab/show/function+application

A function f is defined by its association to each input value x (belonging to some allowable domain of values) of an output value, usually denoted f(x) or fx. The process of passing from f and x to f(x) is called function application, and one speaks of applying f to x to produce f(x).

https://ncatlab.org/nlab/show/Set

Is an identity functor in Category of sets a function application?

The reason I ask this is in proguramming such as F#, pipeline operator

https://riptutorial.com/fsharp/example/14158/pipe-forward-and-backward

 "Hello World" |> print

 value |> f

Now,

 value |> map(f)

is generally recognized as functor.

In this understanding, a simple function application

 value |> f

should be an identity functor, is this correct?

Thanks.

EDIT

(endo)Functor

value |> map(f)

identityFunctor (special case: map == identity)

value |> identity(f)

Therefore, identityFunctor is equivalent to

function application

value |> f 

in another notation,

f(value)

Answer 1

When translating from category theory to a programming language, replace "object" with "type" and "morphism" with "function".

When you say that value |> map(f) is recognized as functor, the functor part is actually related to the type of the value. This type was created by applying a type constructor to some other type. For instance, value may be a list of integers: the functor "list" was applied to "integer". Function f, in this case, operates on integers, but map(f) operates on lists of integers. We say that map "lifts" f to operate on lists.

Identity functor maps every type to itself. So, for instance, it maps type int to type int. In this case map(f) is the same as f: lifted f is again f.

Function application is a morphism. It takes a pair (a product) of the function type and the argument type and maps it to the result type. Here, it takes the pair (f, value).