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A tiny category.
Recall that a category consists of the data:
a set \(Ob(C) \) of objects;
for every pair of objects \(c,d \in Ob(C) \) a set \(C(c,d) \) of morphisms;
for every three objects \(b,c,d \) and morphisms \(f : b \to c \) and \(g : c \to d \), a specified morphism \((f ; g) : b \to d \) called the composite of \(f \) and \(g \);
for every object \(c \), an identity morphism \(id_c \in C(c,c) \); and
subject to two laws:
Unit: for any \(f : c \to d \), the equations \(id_c ; f = f \) and \(f ; id_d = f \) hold.
Associative: for any \(f_1 : c_1 \to c_2, f_2 : c_2 \to c_3 \), and \(f_3 : c_3 \to c_4 \), the equation \((f_1 ; f_2) ; f_3 = f_1 ; (f_2 ; f_3) \) holds.
This tiny category is sometimes called the walking arrow category \(2 \).
(a) Write down the set of objects, the four sets of morphisms, the composition rule, and the identity morphisms.
(b) Prove that this category obeys the unit and associative laws.