# Initial and Terminal Objects

### Definition - Initial Objects

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 $\mathcal{C}$.

### Definition - Terminal Objects

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.

### Example - Set

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 $x$.