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.