In Lurie's "Higher topos theory" lemma 4.3.2.7 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 $F_0$.

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 $F_0$.”?
 A: Let me answer first the question about the homotopy pushout. It is in general not the case. Consider for instance the inclusion $\Delta^0 \to \Delta^1$. Then $\Delta^0 \star * = \Delta^1$ so that the homotopy pushout with $\Delta^1$ along $\Delta^0$ (computed as sinply the pushout, because it's along a cofibration) is just $\Lambda^2_0$, which is already a quasicategory, whereas $\Delta^1\star * = \Delta^2$, and they are not equivalent.
For the question about this comment, note that you can reformulate the statement of the lemma as : the following square is a pullback square :
$\require{AMScd}
\begin{CD}D_{F/} @>>> D_{F_0/} \\
@VVV @VVV \\
D'_{pF/} @>>> D'_{pF_0/} \end{CD}$
Now in this kind of situation, a diagram in the top left hand corner which is $p$-initial when moved to the right was $p$-initial to begin with. This is 4.3.1.5.(4) in HTT.
This shows that a cocone under $F$ wich restricts to a $p$-colimit of $F_0$ is a $p$-colimit of $F$. See Daniel's comment below to see why there shouldn't be a converse in general.
Given a $p$-colimit of $F_0$, say $\overline{F_0}: C_0^\triangleright \to D$, by relative Kan extension, if you're also given an extension $\overline F  : C^\triangleright \to D'$ of $p\circ \overline{F_0}$, then you get a diagram $\Delta^0\to D_{F/}$ which becomes a $p$-colimit of $F_0$ under restriction, so again by the above, a $p$-colimit of $F$.
