Remark 8.4.1.24. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a dense diagram. Choose another diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ and set $\widetilde{\operatorname{\mathcal{D}}} = \operatorname{\mathcal{D}}_{/q}$, so that the projection map $\pi : \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ is a right fibration. Then the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is dense. To prove this, it will suffice to show that for every object $Y \in \widetilde{\operatorname{\mathcal{D}}}$, the induced map
is a colimit diagram in $\widetilde{\operatorname{\mathcal{D}}}$ (Remark 8.4.1.20). By virtue of Proposition 7.1.3.19, this is equivalent to the requirement that $\pi \circ \theta $ is a colimit diagram in $\operatorname{\mathcal{D}}$. Unwinding the definitions, we see that $\pi \circ \theta $ is given by the composition
Since $\pi $ is a right fibration, the map $\widetilde{\operatorname{\mathcal{D}}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }$ is a trivial Kan fibration (Proposition 4.3.7.12). Using Corollary 7.2.2.2, we are reduced to showing that the map $(\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \pi (Y) })^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram, which follows from our assumption that $F$ is dense (Remark 8.4.1.20).