Remark 6.2.4.6. In the situation of Proposition 6.2.4.2, let $w: X \rightarrow Z$ be any morphism in the $\infty $-category $\operatorname{\mathcal{E}}$ and set $C = U(X)$. Then $u$ admits an (essentially unique) factorization
where $v$ is $U$-cartesian and $u$ is a morphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (see Remark 5.1.3.8). In this case, $w$ is a $\operatorname{\mathcal{E}}'$-local equivalence if and only if the following conditions are satisfied:
The morphism $v: Y \rightarrow Z$ is an isomorphism in $\operatorname{\mathcal{E}}$ (since $v$ is $U$-cartesian, this is equivalent to the requirement that $U(v)$ is an isomorphism in $\operatorname{\mathcal{C}}$).
The morphism $u$ is a $\operatorname{\mathcal{E}}'_{C}$-local equivalence in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.