Implication Details

Assumptions: semi-strongly connected, thin

Conclusions: locally cartesian closed

Reason: 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 $b$ .