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An object \( 0 \) in a category \( \mathcal{C} \) is called initial if it has exactly one outgoing arrow from it into every object, \( A \), in \( \mathcal{C} \).
An object \( 1 \) in a category \( \mathcal{C} \) is called terminal if it has exactly one incoming arrow from every object, \( A \), in \( \mathcal{C} \).
We often label these arrows into terminal objects or out from initial objects by ! to highlight their uniqueness.
In \( Set \) the only initial object is the empty set \( \emptyset \). Each one element set, \( \{x\} \), is terminal since there is a function for some arbitrary set, \( S \), that maps every element to \( x \).
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