First, S is the category of sets, whose objects and morphisms are sets and functions, respectively. Next, a functor
P: V –> W
is an assignment of objects and morphisms of the codomain category W to objects and morphisms, respectively, of the domain category V in a way respectful of the domain, codomain, identity, and composition structure of the category.
Let’s look at the case of the functor of our present concern, which has the category S of sets as both domain and codomain category i.e. a functor
P: S –> S
The functor P assigns sets to sets, which we denote as
POb: SOb –> SOb
and functions to functions, which we denote as
PMp: SMp –> SMp
These two functions
POb: SOb –> SOb
PMp: SMp –> SMp
together are required to satisfy, in order to constitute a functor
P: S –> S
the following four conditions corresponding to preserving (i) domain, (ii) codomain, (iii) identity, and (iv) composition.
(i) Preserving Domain
SOb –POb–> SOb
^ ^
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domain domain
| |
SMp –PMp–> SMp
The commutativity of the above diagram stated as
domain o PMp = POb o domain
guarantees the preserving of domain (where ‘o’ denotes composition). This commutativity equation is read as: the domain set (in top-right SOb) of the function (in bottom-right SMp) to which a function (in bottom-left SMp) is assigned to by PMp is same as the set (in top-right SOb) to which the domain set (in top-left SOb), of the function (in bottom-left SMp), is assigned to by POb.
(ii) Preserving Codomain
SOb –POb–> SOb
^ ^
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codomain codomain
| |
SMp –PMp–> SMp
The commutativity of the above diagram stated as
codomain o PMp = POb o codomain
guarantees the preserving of codomain. This commutativity equation is read as: the codomain set (in top-right SOb) of the function (in bottom-right SMp) to which a function (in bottom-left SMp) is assigned to by PMp is same as the set (in top-right SOb) to which the codomain set (in top-left SOb), of the function (in bottom-left SMp), is assigned to by POb.
(iii) Preserving Identity
SMp –PMp–> SMp
^ ^
| |
identity identity
| |
SOb –POb–> SOb
The commutativity of the above diagram stated as
identity o POb = PMp o identity
guarantees the preserving of identity. This commutativity equation is read as: the identity function (in top-right SMp) of the set (in bottom-right SOb) to which a set (in bottom-left SOb) is assigned to by POb is same as the function (in top-right SMp) to which the identity function (in top-left SMp), of the set (in bottom-left SOb), is assigned to by PMp.
(iv) Preserving Composition
PMp (g o f) = PMp (g) o PMp (f)
The function (PMp (g o f)) to which the composite function (g o f) is assigned to by PMp is same as the composite of the functions (PMp(g) and PMp(f)) to which the functions (g and f, respectively) are assigned to by PMp.
Now that we know what it takes to be a functor, let’s see if what we are given i.e. the functions
(-)T(A) = AT
(-)T(f) = f T
together constitute a functor
(-)T: S –> S
The function
(-)TOb: SOb –> SOb
assigns to each set
A
(in the domain set SOb) the map set
AT
of T-shaped figures in A i.e. the functions
a: T –> A
from [a fixed set] T to A.
The function
(-)TMp: SMp –> SMp
assigns to each function
f: A –> B
(in the domain set SMp) the induced function
f T: AT –> BT
where AT and BT are map sets whose elements are T-shaped figures in A and B, respectively.
The function
f T: AT –> BT
assigns to each element (a T-shaped figure in A)
a: T –> A
in the domain map set
AT
the element
T —a–> A —f–> B
i.e. a T-shaped figure in B
fa: T –> B
in the codomain set
BT
Thus
f T(a) = fa
for all
a: T –> A
in the domain map set
AT
of the function
f T: AT –> BT
Now we have to check to see if the object function
(-)TOb (A) = AT
and the morphism function
(-)TMp (f: A –> B) = f T: AT –> BT
together constitute a functor
(-)T: S –> S
i.e. preserve (i) domain, (ii) codomain, (iii) identity, and (iv) composition.
(i) Preserving Domain
SOb –(-)TOb–> SOb
^ ^
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domain domain
| |
SMp –(-)TMp–> SMp
domain o (-)TMp = (-)TOb o domain
LHS
domain o (-)TMp (f: A –> B) = domain (f T: AT –> BT) = AT
RHS
(-)TOb o domain (f: A –> B) = (-)TOb (A) = AT
(ii) Preserving Codomain
SOb –(-)TOb–> SOb
^ ^
| |
codomain codomain
| |
SMp –(-)TMp–> SMp
codomain o (-)TMp = (-)TOb o codomain
LHS
codomain o (-)TMp (f: A –> B) = codomain (f T: AT –> BT) = BT
RHS
(-)TOb o codomain (f: A –> B) = (-)TOb (B) = BT
(iii) Preserving Identity
SMp –(-)TMp–> SMp
^ ^
| |
identity identity
| |
SOb –(-)TOb–> SOb
identity o (-)TOb = (-)TMp o identity
LHS
identity o (-)TOb (A) = identity (AT) = 1AT: AT –> AT
RHS
(-)TMp o identity (A) = (-)TMp (1A: A –> A) = 1AT: AT –> AT
where
1AT (a: T –> A) = T —a–> A –1A–> A = 1A o a = a: T –> A
(iv) Preserving Composition
(-)TMp (g o f) = (-)TMp (g) o (-)TMp (f)
where
f: A –> B, g: B –> C, and g o f: A –> C
LHS
(-)TMp (g o f: A –> C) = (gf)T: AT –> CT
where
(gf)T(a: T –> A) = T —a–> A —gf–> C = (gf)a: T –> C
RHS
(-)TMp (g) o (-)TMp (f)
= gT: BT –> CT o f T: AT –> BT
= AT —f T–> BT —gT–> CT
= gT o f T: AT –> CT
where
gT o f T (a: T –> A) = gT (fa: T –> B) = g(fa): T –> C
Thanks to the associative law
of composition
T —a–> A —f–> B —g–> C
the morphism component
(-)TMp: SMp –> SMp
of the functor
(-)T: S –> S
preserves composition. Thus with (i) domain, (ii) codomain, (iii) identity, and (iv) composition preserved by the assignments
(-)T(A) = AT
(-)T(f) = f T
we do have a functor
(-)T: S –> S