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## Elementary Category Theory - 3

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Tsakanikas Nickos
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### Elementary Category Theory - 3

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 $K = Ker(f)$.