Categories

Categories are constructs that contain the following data:

1. A collection of objects, ${\mathcal{C}}_0$ (like a set but a collection of all sets also contains the set of all sets)
2. For any two objects $X,Y\in {\mathcal{C}}_0$, a collection of morphisms from $X$ to $Y$. This is denoted by: ${\text{hom}}_C(X,Y)$, $C(X,Y)$, $X\to Y$.
3. For each object $X\in \mathcal{C}$, a morphism ${\text{Id}}_X\in C(X,X)$ that maps $X$ to itself.
4. A binary operation called composition ${\circ }_{(}X,Y,Z)$ that can join morphisms together such that ${\circ }_{(}X,Y,Z):{\text{hom}}_C(X,Y)\to {\text{hom}}_C(Y,Z)\to {\text{hom}}_C(X,Z)$

The properties should be satisfied for this to be considered a category:

1. Left Unit Law: $f\in C(X,Y)$ , $f\circ {\text{Id}}_X=f$
2. Right Unit Law: $f\in C(X,Y)$ , ${\text{Id}}_X\circ f=f$
3. Associative Law: $h\circ (g\circ f)=(h\circ g)\circ f$

Sets Category

The category of sets, denoted by $\text{Set}$ is the category such that:

1. An object is a set
2. Given two sets $X,Y$ $\text{Set}(X,Y)$ is the set of all functions from $X$ to $Y$
3. ${\text{Id}}_X$ is the identity function on the set $X$
4. Composition operation is just function compositions

Additonal Steps

To prove set is a category, final step is to show that the left/right unit laws and the associative law holds.