An initial object of a category has exactly one morphism for every object . 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, 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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