Corollary 9.1.4.9. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category, let $(Q, \leq )$ be a directed partially ordered set, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ be a functor. Then $F$ is right cofinal if and only if it is weakly right cofinal.
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Proof. Assume that $F$ is weakly right cofinal; we will show that $F$ satisfies the criterion of Remark 9.1.4.6, and is therefore right cofinal (the converse follows from Example 9.1.4.2). Fix an object $X \in \operatorname{\mathcal{C}}$ and an element $q \in Q$ satisfying $F(X) \leq q$. We wish to show that there exists a morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ such that $q \leq F(Z)$. Applying our assumption that $F$ is weakly right cofinal, we can choose an object $Y \in \operatorname{\mathcal{C}}$ satisfying $q \leq F( Y )$. We complete the proof by observing that there is another object $Z \in \operatorname{\mathcal{C}}$ with morphisms $f: X \rightarrow Z$ and $g: Y \rightarrow Z$, by virtue of our assumption that $\operatorname{\mathcal{C}}$ is filtered. $\square$