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An initial object in a category
 is an object  in
 such that, for every object  in
 , there is exactly one morphism
 .
A terminal object in a category
 is an object  in
 such that, for every object  in
 , there is exactly one morphism
 .
A zero object in a category
 is an object 0 that is both an initial object and a terminal object.
All initial objects (respectively, terminal objects, and zero objects), if they exist, are isomorphic in
 .
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