= Initial object
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An <initial object> $0$ of a <category> has exactly one <morphism> $0\to X$ for every object $X$. Any two initial objects are uniquely isomorphic: the unique maps in both directions compose to the unique endomorphisms, which must be identities.
The empty <set> is initial in the <Category of sets>, since there is one empty <function> to any set. In the category of unital <commutative rings>, $\mathbb Z$ is initial, as the image of one determines the whole homomorphism. Initiality concerns outgoing maps; a <terminal object> has the dual incoming-map property.
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