Let's say that an arrow $ f\colon X\to Y $ in a category $ \mathcal C $ have an image if there exists a mono $ \iota\colon \operatorname{Im} f\rightarrowtail Y $ through which $ f $ factors, such that for every other mono $ i\colon I\rightarrowtail Y $ with this property there exists a unique morphism of $ \mathcal C/Y $ from $ \iota $ to $ i $.
Moreover, call a cover any arrow $ \phi\colon U\to X $ such that any mono $ V\rightarrowtail X $ through which $ \phi $ factors is an isomorphism.
Johnstone in its Elephant claims that in a finitely complete category images are stable under pullback if and only if covers are. In other words, given $ f_1\colon X_1\to Y $ and $ f_2\colon X_2\to Y $, then $ f_2^*(\operatorname{Im}f_1)\cong \operatorname{Im}{f_2^*(f_1)} $ in $ \mathcal C/{X_2} $ if an only if the pullback of a cover is again a cover.
I have two questions:
- I couldn't manage to prove this funny fact. Could someone help me?
- Why we want images to be stable under pullback? Could someone provide me an example?
The really important questions is obviously pt. 1. By the way I know that I could answer pt. 2 simply by continuing reading, but having some intuition before trying to digest 1400 pages of math is always desirable.