The question here explains how a cartesian product of sets are specified. It is difficult for me to follow it; for example, I do not know about "symmetric monoidal category".
Can anybody please explain cartesian product construction using simpler concepts? for example, what set of constraint should be checked to see if an object is a cartesian product of another two objects without looking inside the objects?
PS: Question edited to avoid duplication
Suppose that $P : \mathsf{Rel} \times \mathsf{Rel} \to \mathsf{Rel}$ is a functor equipped with natural isomorphisms $$\hom(P(X,Y),Z) \cong \hom(X,P(Y,Z))$$ and $$P(X,1) \cong X \cong P(X,1).$$
Then $P(-,Y)$ is left adjoint to $P(Y,-)$, so that it preserves colimits, in particular coproducts. This implies $$P(X,Y) \cong P(\coprod_{x \in X} 1,Y) \cong \coprod_{x \in X} P(1,Y) \cong \coprod_{x \in X} Y \cong X \times Y.$$