Definition:Composition of Mappings/Commutative Diagram

Definition
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}$