Remark 7.1.9.6. In the situation of Proposition 7.1.9.4, suppose that the diagram $q$ factors through the homotopy fiber product
that is, it corresponds to a triple $(q_0, q_1, \alpha )$ where $\alpha : F_0 \circ q_0 \rightarrow F_1 \circ q_1$ is an isomorphism of $K$-indexed diagrams in $\operatorname{\mathcal{C}}$. In this case, the following conditions are equivalent:
The colimit of $q$ (formed in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$) also belongs to the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.
The natural transformation $\overline{\alpha }: F_0 \circ \overline{q}_0 \rightarrow F_1 \circ \overline{q}_1$ appearing in the proof of Proposition 7.1.9.4 is an isomorphism.
The composition $F_1 \circ \overline{q}_1$ is a colimit diagram in $\operatorname{\mathcal{C}}$: that is, the functor $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ preserves the colimit of the diagram $q_1$.
See Proposition 7.1.3.18.