Implication Details
Claim: Given a symmetric monoidal category whose underlying category is infinitary distributive, if it is cartesian, then it is infinitary distributive.
Proof: This is just the definition of an infinitary distributive category.
Claim: Given a symmetric monoidal category whose underlying category is infinitary distributive, if it is cartesian, then it is infinitary distributive.
Proof: This is just the definition of an infinitary distributive category.