Is the product of two objectos $a,b$ in a preorder category their infimum $a\wedge b$? I can't assume that their infimum exists, can I?
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Yes, the product of $a$ and $b$ is $a\land b$, if it exists; however, this is the infimum, not the supremum. As you suspect, it need not exist for arbitrary $a$ and $b$.