Warning 4.8.8.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. In the case $n = 0$, our proof of Theorem 4.8.8.3 shows that $F$ factors as a composition
where $F'$ is fully faithful and essentially surjective, and $G$ is a $0$-categorical inner fibration (in particular, $G$ is faithful). Beware that generally does not coincide with the factorization constructed in Example 4.8.8.6. If $u: X \rightarrow Y$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ having the property that $F(u)$ is an identity morphism in $\operatorname{\mathcal{D}}$, then the functor $F'$ carries $X$ and $Y$ to the same object of $\mathrm{h}_{\mathit{\leq 0}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. Consequently, the functor $F'$ is generally not bijective on objects.
A related phenomenon occurs in the case $n = -1$. By construction, $\mathrm{h}_{\mathit{\leq -1}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ is the full subcategory of $\operatorname{\mathcal{D}}$ spanned by objects of the form $F(X)$, where $X$ is an object of $\operatorname{\mathcal{C}}$. If the inner fibration $U$ is not an isofibration, this subcategory might be smaller than the essential image of $F$.