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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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