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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