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Assume $\mathbb A$ is category. Let $h:A \rightarrow A$ be an arrow in $\mathbb A$. I want to know if $h$ is the identity on $A$. Is this true if $\forall X \in obj(\mathbb A) \forall f:A \rightarrow X \in hom(A,X) : f \circ h = f$ ?

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Do you mean $f\circ h=f$? –  Stefan Hamcke Sep 10 '13 at 16:18
    
Sorry, that was a typo. Corrected it. –  André Sep 10 '13 at 16:21
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1 Answer

up vote 2 down vote accepted

You probably meant $f \circ h = f$, not $f \circ h = h$. And the answer is yes: if that condition holds, taking $f = id_A$ you get $id_A \circ h = id_A$, so $h = id_A$.

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Sure, yes. I am new to this so I want to make sure I get everything right. Thanks. –  André Sep 10 '13 at 16:22
    
Omar's demonstration is typical in these kind of proofs: identities (or units or neutral elements) are uniquely determined by their defining equations, if they are already known to exist –  magma Sep 10 '13 at 23:25
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