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The Concept of Functor
in Category Theory

A category is a collection of objects and a collection of arrows between those object. The collection of arrows is closed
under composition. A functor is mapping between two categories. Because any cateory involves two collections,
objects and arrows, a functor has to involve two mappings; one mapping between the object collections of the two categories
and another between the arrow collections.

The functor mappings have satisfy further conditions. Let the two categories for a functor be denoted as C and D and a functor
between them as F:C → D. Just for this presentation think of C and D as the order pairs (C_{O}, C_{A})
and (D_{O}, D_{A}). Likewise F is an ordered pair (F_{O}, F_{A}) where

if f ∈ F_{O} then f:C_{O} → D_{O} and
if φ ∈ F_{A} then φ:C_{A} → D_{A}

The conditions which F must satisfy are:

(i) If φ ∈ C_{A} and φ:O_{1>}→O_{2}
then the ψ ∈ D_{A} such that ψ:F_{O}(O_{1})→F_{O}(O_{2})
is F_{A}(φ)
(ii) F_{A}(f o g) = F_{A}(f) o F_{A}(g)
(iii) For any object O in C:
F_{O}(1_{O1}) = 1_{FO(O1)
}

The remarable and beautiful thing is that the collection of categories with functors as the collection of arrows is a CATEGORY!