# Proof that Rel is a Category

The Awodey book about Category Theory gives this definition for Rel:

The objects of $\text{Rel}$ are sets, and an arrow $A → B$ is a relation from $A$ to $B$, that is, a subset $R ⊆ A×B$.

The equality relation $\{ \langle a, a\rangle ∈ A×A\;|\; a ∈ A\}$ is the identity arrow on a set $A$.

Composition in $\text{Rel}$ is to be given by: $$S ◦ R = \{\langle a, c \rangle ∈ A × C \; |\; ∃b \;( \langle a, b\rangle ∈ R \; \text{ and }\; \langle b, c\rangle ∈ S) \}$$

for $R ⊆ A × B$ and $S ⊆ B × C$.

How can I actually prove that Rel is a category?

I'm not really into categories yet but this seem quite obvious since I know a category must have objects (that are the sets, in this case), arrows (relations), identity function (equality relation) and composition of functions (that is the association of relations).