Implication details

Proof: Let $D : \I \to \C$ be a cosifted diagram in a direct category $\C$. Assume that for every $i \in \I$ there is a morphism $j \to i$ in $\I$ such that $D(j) \to D(i)$ is not an identity. Since $\I$ is non-empty, this yields an infinite sequence of non-identity morphisms $\cdots \to D(i_2) \to D(i_1) \to D(i_0)$, a contradiction. Thus, there is some $i_0 \in \I$ such that $D(j) \to D(i_0)$ is an identity for every morphism $j \to i_0$. By the dual of Lemma 1 here, $D(i_0)$ is a limit object of $D$.