Remark 6.2.4.3. 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 $C = U(Z)$. Then $w$ admits an (essentially unique) factorization
where $u$ is $U$-cocartesian and $v$ 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 $u: X \rightarrow X'$ is an isomorphism in $\operatorname{\mathcal{E}}$ (since $u$ is $U$-cocartesian, this is equivalent to the requirement that $U(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$: see Proposition 5.1.1.9).
The morphism $v$ is a $\operatorname{\mathcal{E}}'_{C}$-local equivalence in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.