Implication Details

Claim: Given a functor whose codomain is thin, and whose domain is semi-strongly connected and is thin, then it preserves binary products.

Proof: It is enough to prove that if $P$ is a linearly ordered set and $Q$ is a partially ordered set, then any order-preserving map $f : P \to Q$ preserves binary meets. Let $x,y \in P$. We may assume $x \leq y$. Then $x \wedge y = x$. Since we also have $f(x) \leq f(y)$, we have $f(x) \wedge f(y) = f(x)$. Therefore, $f(x \wedge y) = f(x) = f(x) \wedge f(y)$ holds.

This implication has a dual.