Examples of Functors
  1. Identity functor - for a category X an identity X → X keeps objects snd arrows intact. Still it is a functor.
  2. Setf Set - a functor that includes Setf into Set, that is, maps each finite set to itself, and the same with functions. Note that this is not an identity functor.
  3. SetTop - similar to the previous example, this functor makes Set a part of Top. Each set is mapped to a discrete topological space.
  4. For any set A we can define the following functor:
    (- xA): SetSet - it will map any set X to a cartesian product, X × A.
  5. For any set A we can define a functor
    PA: SetSet; it maps any set X to XA, a set of functions from A to X.
  6. Set Part embeds sets to sets with partial functions: it maps sets and functions to themselves.
  7. An opposite to 6. is a functor +Null: PartSet -this functor adds an "extension" Null to each set: X ↦ (X+Null), so that a partial function X → Y maps to a function (X+Null) → (Y+Null).
    (Exercise. Define such an extension for partial functions.)