Implication Details
Claim: Given a symmetric monoidal category whose underlying category is cartesian closed, if it is cartesian, then it is closed.
Proof: This is just the definition of a cartesian closed category.
This implication has a dual.
Claim: Given a symmetric monoidal category whose underlying category is cartesian closed, if it is cartesian, then it is closed.
Proof: This is just the definition of a cartesian closed category.
This implication has a dual.