Let \(x,y \in P\) be elements of a preorder and \(\mathcal{P}\) be the corresponding category. Show that the product \(x \times y\) in \(\mathcal{P}\) agrees with their meet \(x \land y\) in \(P\).
The uniqueness aspect of the product is not relevant since all morphisms are unique in a preorder category.
The product definition translates to an operation which takes a pair of objects in a preorder and gives us another object with the property that \(x \times y \leq x\) and \(x \times y \leq y\), and for any other b that also has this property we have \(b \leq x\times y\)
Considering the set \(A=\{x,y\}\), the conditions for \(x \times y\) matches the definition of \(\bigwedge A\) (grestest lower bound).