\begin{definition}[Definition:Composition of Mappings/Commutative Diagram]
Let $S_1$, $S_2$ and $S_3$ be sets.
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.
The concept of '''composition of mappings''' can be illustrated by means of a commutative diagram.
This diagram illustrates the specific example of $f_2 \circ f_1$:
::$\begin{xy}\xymatrix@+1em{
 S_1
  \ar[r]^*+{f_1}
  \ar@{-->}[rd]_*[l]+{f_2 \mathop \circ f_1}
&
 S_2
  \ar[d]^*+{f_2}
\\
&
 S_3
}\end{xy}$
\end{definition}