Definition 9.2.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We say that an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local if, for every morphism $f: X \rightarrow C$, there exists a $2$-simplex with boundary indicated in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar@ {-->}[dr] & \\ X \ar [ur]^{w} \ar [rr]^{f} & & C. } \]
If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local if it is weakly $w$-local for each $w \in W$.