In Lurie's "[Higher topos theory](https://arxiv.org/abs/math/0608040)" 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
> [![\xymatrix{ \calC^{0} \ar@{^{(}->}[d] \ar[r]^{F_0} & \calD \ar[d]^{p} \\
\calC \ar[r] \ar[ur]^{F} & \calD' }][1]][1]
>
> 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 $F_0$.

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 $F_0$.”?


  [1]: https://i.sstatic.net/fz1EJ.png