Implication details

Proof: Let $\C$ be an essentially small direct category. The proof of this implication shows that for every cosifted diagram $D : \I \to \C$ there is an object $i_0 \in \I$ such that $D(a)$ is an isomorphism (even an identity) for every morphism $a : i \to i_0$ in $\I$. In other words, the diagram $D^{\op} : \I^{\op} \to \C^{\op}$ is eventually constant in the sense of this page. Therefore, the claim follows from the duals of Corollaries 3 and 4 there.