In Lurie's "Higher topos theory" lemma 4.3.2.7, I’m trying to understand “In particular, we may identify $p$-colimits of $F$ with $p$-colimits of $F_0$”:
Lemma 4.3.2.7. Suppose we are given a diagram of $\infty$-categories
as in Definition 4.3.2.2, where $p$ is a categorical fibration and $F$ is a $p$-left Kan extension of $F_0$. Then the induced map $$\mathcal D_{F/} \to \mathcal D'_{p F/} \times_{\mathcal D'_{p F_0/}} \mathcal D_{F_0/}$$ is a trivial fibration of simplicial sets. In particular, we may identify $p$-colimits of $F$ with $p$-colimits of $F_0$.
![\xymatrix{ \calC^{0} \ar@{^{(}->}[d] \ar[r]^{F_0} & \calD \ar[d]^{p} \
\calC \ar[r] \ar[ur]^{F} & \calD' }](https://i.sstatic.net/fz1EJ.png)
It seems that here we have $C\rightarrow C*\mathrm{pt}$ is a homotopy pushout of right cone $C_0\rightarrow C_0*\mathrm{pt}$ via the embedding of quasi-categories $C_0\rightarrow C$, but why?
Or, is there another way to figure out “In particular, we may identify $p$-colimits of $F$ with $p$-colimits of $F_0$.”?
