This is a theorem due to Thomas Fox which characterizes products in a cartesian monoidal category as an algebraic structure given by natural transformations rather than in terms of a universal poperty.
A symmetric monoidal category is cartesian if and only if it is isomorphic to its own category of cocommutative comonoids. Thus every object is equipped with a unique cocommutative comonoid structure, and these structures are respected by all maps.
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