General Definition
Let be a category and let be a class of morphisms of .
An object of is said to be -injective if for every arrow and every morphism in there exists a morphism extending (the domain of), i.e . In other words, is injective iff any -morphism extends (via composition on the left) to any morphism into .
The morphism in the above definition is not required to be uniquely determined by .
In a locally small category, it is equivalent to require that the hom functor carries -morphisms to epimorphisms (surjections).
The classical choice for is the class of monomorphisms, in this case, the expression injective object is used.
Read more about this topic: Injective Object
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