According to Wikipedia, image of a morphism $\phi:X\rightarrow Y$ in a category is a monomorphism $i:I\rightarrow Y$ satisfying the following conditions:
- There is a morphism $\alpha:X\rightarrow I$ such that $i\circ\alpha=\phi$.
- If $j:J\rightarrow Y$ is a monomorphism and $\beta:X\rightarrow J$ is a morphism such that $\beta\circ j=\phi$, then there exists a unique morphism $\gamma:I\rightarrow J$ such that $\beta=\gamma\circ\alpha$ and $j\circ\gamma=i$.
It is easy to see that $\alpha$ is unique. Intuitively, such $\alpha$ should be an epimorphism, but I can't seem to show it. Is it true?