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We know that product in slice category $\mathcal{C}\downarrow x$ is pullback in $\mathcal{C}$, but what's a coproduct in $\mathcal{C}\downarrow x$ (described in $\mathcal{C}$)? I tried to picture it as a limit or colimit, without success.

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The coproduct of $y \to x$ and $z \to x$ is $y \sqcup z \to x$. (Note that the forgetful functor from the slice category back down to $C$ has a right adjoint whenever $C$ has all products by $x$, and in that case it preserves all colimits. But actually it just always preserves all colimits.)

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Is $\sqcup$ here a standard notation? How's it pronounced? – Henning Makholm Jan 24 '15 at 4:03
@Henning: I don't know how to pronounce it. It's reasonably standardish notation for coproducts, e.g. it's the notation used by both Wikipedia and the nLab. – Qiaochu Yuan Jan 24 '15 at 6:06
Ah, I see. So the answer is that if $y$ and $z$ have a coproduct in $\mathcal C$, then that coproduct induces a coproduct in the slice category? But if not, then the two arrows could still have a coproduct in the slice category. – Henning Makholm Jan 24 '15 at 12:22

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