Commutative Diagram/Examples/Arbitrary Example 1

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Example of Commutative Diagram

Let $A, B, X, Y$ be arbitrary sets.


\(\ds f: \, \) \(\ds A\) \(\to\) \(\ds X\)
\(\ds g: \, \) \(\ds B\) \(\to\) \(\ds Y\)
\(\ds \alpha: \, \) \(\ds A\) \(\to\) \(\ds B\)
\(\ds \beta: \, \) \(\ds X\) \(\to\) \(\ds Y\)

be mappings such that:

$\beta \circ f = g \circ \alpha = k$

where $\circ$ denotes composition of mappings.

This can be depicted using the following commutative diagram:

$\quad\quad \begin{xy} \xymatrix@L+2mu@+1em{ A \ar[r]^*{\alpha} \ar[d]_*{f} \ar[rd]^*{k} & B \ar[d]^*{g} \\ X \ar[r]^*{\beta} & Y }\end{xy}$