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PostPosted: Sun Mar 06, 2016 2:50 pm 
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Let \( \mathcal{C} \) be an abelian category and let \( f \ \colon X \longrightarrow Y \) be a morphism in \( \mathcal{C} \). Show that the diagram
\[ \xymatrix{ K \ar[d] \ar[r]^{0_{K}} & 0 \ar[d] \\ X \ar[r]_f &Y} \]
is a pull-back diagram if and only if \( K = Ker(f) \).


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