In the beginning, it is mentioned that a category-theoretical definition may supersede a componential definition. In this section:

1. We first observe a definition of the ‘set’ in the category theory language and realize the necessity of the distinction between the set and the non-set.
2. Using this fact, we categorize categories by whether the size is small enough to consider as a set or not. This is necessary since the later application of category theory will make use of the category $\text{Set}$ quite often.
3. The last will compare with the other definitions of ‘set’, which justify the opinion that this approach takes a closer position to our intuition than other traditional definitions.

Constructions with Sets

Our intuition says that the set is ‘a bag of featureless points’, and the function is ‘an assignment of the points from one to another, and the composition is just a sequential assignment. These make $\textbf{Set}$ a category, but it lacks a precise explanation of ‘what it is’. To resolve it, we list several categorical properties of the category that can be derived from our intuition:

• $\emptyset$(empty set) - initial object
• $1$(one-element set) - terminal object, represents elements
• $A\times B$, $\prod_{i\in I}A_i$(product) - product
• $A+B$, $\sum_{i\in I}A_i$(sum/disjoint union) - coproduct
• $B^A$(set of functions) - $\prod_{a\in A}B$
• Digression on arithmetic - cartesian closed
• $2$(two-element set) - $1+1$, represents subsets(i.e. subobject classifier)
• $\chi_S :A→2$(characteristic function ↔ subset)
• $\mathcal{P}(A)=2^A$(power set)
• $\{a\in A:f(a)=g(a)\}$(equalizer) - terminal fork
• $A/\sim$(quotient) - universal property
• $\mathbb{N}$(natural number), $X^\mathbb{N}$(sequence) - universal property of the mathematical induction system
• section/right inverse of $f$ - $f\circ i=1_B$
• axiom of choice - surjective⇒$\exists$section