Implication Details
Claim: If a category is cartesian closed and has filtered colimits, then it has cartesian filtered colimits.
Proof: Each functor is a left adjoint and therefore preserves (filtered) colimits.
Claim: If a category is cartesian closed and has filtered colimits, then it has cartesian filtered colimits.
Proof: Each functor is a left adjoint and therefore preserves (filtered) colimits.