The composition of two binary relations $R$ and $S$ is the binary relation
\[S \circ R = \{(x, z) : (\exists y : x \,R\, y \,\land\, y \,S\, z) \}.\]This is a set by Separation Schema:
\[S \circ R \subseteq \dom R \times \im S.\]
Let $X$ and $Y$ be two sets. For any binary relation $R$ on $X$ and $Y$,
\[R \circ \id_X = \id_Y \circ R = R.\]
For any binary relations $R$, $S$ and $T$,
\[(R \circ S) \circ T = R \circ (S \circ T).\]
For any functions $f$ and $g$ on a set $X$, $f \circ g$ is also a function on $X$.
As a corollary of associativity of $\circ$ and Proposition 1:
For any set $X$, $(\rel(X, X), \circ)$ is a monoid with neutral element $\id_X$.
As a corollary of Proposition 1 and SUB-MOID > Proposition 1:
For any set $X$, $\fun(X, X)$ is a submonoid of $\rel(X, X)$ with neutral element $\id_X$.
For any binary relations $R$ and $S$,
\[(R \circ S)^{-1} = S^{-1} \circ R^{-1}.\]