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I've (partially) read some books about category theory. But only now attempting to put it into research practice I noticed that I do not really understand direct products.

Consider a product $A\times B$ of arrows $x_1: X\rightarrow A$ and $x_2: X\rightarrow B$.

Let now the category Set and $A=\varnothing$. Then there are no arrow $x_1: X\rightarrow A$.

So direct product (in Set) with an empty set does not exist. (I previously though that it exist. Was I wrong?)

I understand something in a wrong way. Please help to understand it properly.

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    I assume you're writing $X$ for $A \times B$. The conclusion has to be that $\emptyset \times B = \emptyset$. Check that this satisfies the universal property!2012-07-22
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    @DylanMoreland: No, $X$ is an arbitrary set.2012-07-22
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    There is an arrow $X \to \emptyset$ if $X = \emptyset$ !2012-07-22
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    @porton I don't see how that shows that $A \times B$ does not exist. The definition of the product of two elements in a category does not say that for any $X$ in the category there exists an arrow $X \to A$. And with $A = \emptyset$ there is only one such $X$ and one such arrow.2012-07-22

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