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Although in $\mathsf{Set}$, canonical projections from a product are surjective and canonical injections to a coproduct are in fact injections, there seems to be nothing forcing this to be the case elsewhere, and indeed Wikipedia indicates without proof that they needn't be epic/monic. Unfortunately, I'm a rank beginner and have not yet knowingly read about a category less rich in monomorphisms and epimorphisms than $\mathsf{Set}$ (though Aluffi has already introduced a couple examples that are richer in them), so I don't know where to try to find a counterexample. Any hints?

To clarify: I am seeking canonical injections to coproducts that are not monomorphisms and/or canonical projections from products that are not epimorphisms.

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Your claim is incorrect: the projection $X \times \emptyset \to X$ is surjective if and only if $X = \emptyset$. –  Zhen Lin Nov 25 '13 at 8:22
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marked as duplicate by Martin Brandenburg, Lord_Farin, AlexR, Zhen Lin, Johannes Kloos Nov 25 '13 at 10:52

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up vote 4 down vote accepted

Take a category that looks like this: $X \leftarrow P \rightarrow Y \rightrightarrows Z$. It has four objects, $X,Y,Z,P$, the only nonidentity morphisms are the ones I indicated (there are four), and the two composites $P \to Y \to Z$ which are the same. $P$ is clearly the product of $X$ and $Y$. But the projection $P \to Y$ is not an epimorphism.

The opposite of this category gives a counterexample for injection.

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Nice and simple, and I hope an approach I can learn from. I will now attempt to come up with a category with all products that is badly behaved in this fashion. Wish me luck. –  dfeuer Nov 25 '13 at 2:36
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