Definition of Injective Object

 Let C be a category.  An object I\in C is injective if we have the following:

English: For every mono m : X \rightarrowtail Y and map f: Y \to I, there exists a map g:Y\to I such that the triangle commutes, or in other words gm = f.

Equivalently, for each mono m: X \rightarrowtail Y the induced map m^*: \text{Hom}(Y, I) \to \text{Hom}(X,I) given by m^*(g) = gm is surjective.  This is actually a nice way to handle the inherent \exists g logic, that I have not considered before.

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