Proving unit relation features

Question

I am trying to prove the following unit relation features:

$R ⊆ X × X$,

$IdX ◦ R = R ◦ IdX = R.$

How do I go about doing it?

Answer 1

$$(x,z)\in\mathsf{Id}_X\circ R\iff \exists y\;[(x,y)\in R\wedge (y,z)\in\mathsf{Id}_X]\tag1$$

Now realize the $(y,z)\in\mathsf{Id}_X$ is true if and only if $y=z$ so $(1)$ can be rewritten as:$$(x,z)\in\mathsf{Id}_X\circ R\iff(x,z)\in R$$

This tells us exactly that: $$\mathsf{Id}_X\circ R=R$$ as was to be shown.

Try to find a similar proof for the other equality.