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I understand why in the category of sets two parallel morphisms $f, g: A \rightarrow B$ are identical iff for each element $x: 1 \rightarrow A$ it holds that $f\circ x = g \circ x$.

Awodey on p. 36 of Category Theory asks (as an exercise), why in any category two parallel morphisms $f, g: A \rightarrow B$ are identical iff for each generalized element $x: X \rightarrow A$ it holds that $f\circ x = g \circ x$.

Could someone please give me a hint how to prove this?

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1 Answer 1

up vote 8 down vote accepted

Just let $X=A$ and $x$ be the identity morphism.

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It's so simple? –  Hans Stricker Sep 9 '11 at 10:09
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@Hans: It couldn't really be anything else, since in general there's no reason for there to be any other morphisms into $A$. –  Chris Eagle Sep 9 '11 at 10:11
    
and the converse? –  user42912 Apr 9 at 9:23

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