Proposition 4.8.5.33. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n \geq 2$. Assume that $G \circ F$ is $n$-full and that $F$ is essentially surjective, full, and $(n-1)$-full. Then $G$ is $n$-full.
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Proof of Proposition 4.8.5.33. Without loss of generality, we may assume that $F$ and $G$ are inner fibrations. Using Proposition 4.8.5.27, we can reduce to the case where $\operatorname{\mathcal{E}}$ is a standard simplex. In this case, we must show that for every morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ of $\operatorname{\mathcal{D}}$, the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ), \overline{u} )$ consists of a single element. Since $F$ is full and essentially surjective, we can assume without loss of generality that $\overline{u} = F(u)$ for some morphism $u: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. In this case, our assumption that $F$ is $(n-1)$-full guarantees that the map
\[ \pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u) \rightarrow \pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}), \overline{u} ) \]
is surjective. It will therefore suffice to show that the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u)$ consists of a single element, which follows from our assumption that $G \circ F$ is $n$-full. $\square$