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Let $i_P:P\rightarrow P\oplus Q$ and $i_Q:Q\rightarrow P\oplus Q$ be a coproduct in an abelain category $\mathcal A$, let $t:T\rightarrow P$,$u:T\rightarrow Q$ be arrows such that $i_Pt=i_Qu$, I have to prove $t=u=0$.

The only thing I came up with was to show that $0$; the initial and terminal pbject of the category, is the pullback of the arrows $i_P$ and $i_Q$, I'm sure this must be true, but how can I prove it?

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up vote 3 down vote accepted

Look at the projection maps $p_P:P\oplus Q \rightarrow P$ and $p_Q:P\oplus Q \rightarrow Q$ (those associated to the product structure of $P\oplus Q$). We have $p_Pi_P = id_P$ and $p_Pi_Q = 0$. (we have similar properties for $P$ and $Q$ switched.) Now just apply this to $i_Pt = i_Qu$.

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I can't use this, since I'm proving that the coproduct is the product, and you're using this fact – Родион Раскольников Jun 2 '13 at 18:26
Ahh, ok. How about this: let $p_P:P\oplus Q \rightarrow P$ that comes from the coproduct universal property characterized by the maps $id:P \rightarrow P$ and $0:Q \rightarrow P$. Define $p_Q$ similarly. Now the above argument after "We have..." all follows. – dc2814 Jun 2 '13 at 18:47

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