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This proposition and its proof use both $F_0$ and $F$ along with $F_1$ and $F'$ as names for the same object.

Yep. Thanks!

The target of $F_0$ and $F_1$ should be $\mathcal{C}$ (not $\mathcal{D}$).

I think $F_0$ (resp. $F_1$) and $\overline{F}_0$ (resp. $\overline{F}_1$) denote the same objects.

I think $H$ should be $F$ (at the end of the second paragraph)

I think the functor $G$ should be the composition of $(\pi, \rm{id})$ and $F$ (as opposed to the projection on $\Delta^1$).

I am confused by the end of the proof. Couldn't I instead say that the functor $F\colon \Delta^1\times \mathcal{K}^{\vartriangleright} \to \mathcal{C}$ induces a functor $F'\colon (\Delta^1\times \mathcal{K})^{\vartriangleright} \to \mathcal{C}$ (as the composition). Then Proposition 030V ensures that $F'$ is a $U$-colimit if and only if the composition $(0\times \mathcal{K})^{\vartriangleright} \to (\Delta^1\times \mathcal{K})^{\vartriangleright} \to \mathcal{C}$ is a $U$-colimit. This composition is equal to $G$, so $G$ is $U$-colimit iff $F'$ is a $U$-colimit. Now $F'$ is a $U$-colimit by Corollary 02NK and Corollary 02XQ.

Sorry, $F_0$ (resp. $F_1$) and $F_0$ (resp. $F_1$) do not denote the same objects. What confused me is that the meaning of $F_0$ (resp. $F_1$) changes within the proof.

Yep. Thanks!