Implication Details
Claim: If a category is cartesian closed and is thin, then it is locally cartesian closed.
Proof: In a thin category, every object is subterminal. Thus, the result follows from Corollary 6 here.
Claim: If a category is cartesian closed and is thin, then it is locally cartesian closed.
Proof: In a thin category, every object is subterminal. Thus, the result follows from Corollary 6 here.