Proof that Rel is a Category

What you've quoted is a definition of all the data needed to have a category: objects, morphisms, identity morphisms, and composition operation. To verify you have a category you then just have to check that this data satisfies the axioms for a category: that the identity morphisms are actually identities for the composition operation, and that composition is associative.

Even though this question was posted quite some time ago I feel like this exercise is very useful for anyone getting into category theory, which is why I provided a solution.

As Eric said, to show that $\mathbf{Rel}$ is a category we need to show that the identity arrows satisfy the identity law and the composition satisfies the associativity law.

Let $f: A \to B$ be a subset $f \subseteq A \times B$, $g: B \to C$ be subset $g \subseteq B \times C$, and $h: C \to D$ be a subset $h \subseteq C \times D$.

To show that $f \circ 1_A = f$, we use the axiom of extensionality (for both are sets). For the forward direction, let $(a, b) \in f \circ 1_A$. Then we see by definition that there exists some $p$ s.t. $(a, p) \in 1_A$ and $(p, b) \in f$. Since $(a, p) \in 1_A$ implies that $p=a$, we have that $(a, b) \in f$. For the backward direction, let $(a, b) \in f$. Moreover, since $(a, a) \in 1_A$, we have per definition that $(a, b) \in f \circ 1_A$. The argument that $1_B \circ f = f$ is similar.

For the associativity, we want to show that $(h \circ g) \circ f = h \circ (g \circ f)$. We know that \begin{align*} (a, b) \in (h \circ g) \circ f &\iff \exists b: (a, b) \in f, (b, d) \in h \circ g \\ &\iff \exists b, c: (a, b) \in f, (b, c) \in g, (c, d) \in h \\ &\iff \exists c: (a, c) \in g \circ f, (c, d) \in h \\ &\iff (a, d) \in h \circ (g \circ f), \end{align*} which is what we wanted to show.