Implication details

Proof: By the proof of this implication, the category has cofiltered limits, and the limit of a cofiltered diagram is one of its objects. We claim that every object $X$ is finitely copresentable. Then the claim follows by taking $G$ to be a collection of representatives of all objects up to isomorphism, which is essentially small by assumption, since every object is trivially a cofiltered limit of objects in $G$.Since the category is thin, $X$ is finitely copresentable if and only if for every cofiltered diagram whose limit $L$ satisfies $L \leq X$, there is an object $Y$ of the diagram with $Y \leq X$. Since $L$ is an object of the diagram, we may take $Y \coloneqq L$.