Implication Details

Claim: If a category is semi-strongly connected and is thin, then it is locally cartesian closed.

Proof: Each slice is thin, semi-strongly connected, and has a terminal object. Thus, it corresponds to a linear order with a largest element $1$. Every such category is cartesian closed, where the exponential $a \Rightarrow b$ (Heyting implication) is $1$ when $a \leq b$ and otherwise $b$.

This implication has a dual.