n-truncation/n-connected factorization in an $\infty$-topoi

Question

I want to prove that given an $\infty$-topos $\mathscr{C}$ and a morphism $f: X \to Y$ in $\mathscr{C}$, for each $k \geq -1$, there exists a factorization $X \xrightarrow{\eta_f} E_k^f \xrightarrow{\tau_{\leq k}(f)} Y$ of $f$ such that $\eta_f$ is $k$-connected and $\tau_{\leq k}(f)$ is $k$-connected. The fact that a factorization exists is guaranteed by the existence of the truncation functors $\tau_{\leq k}$ for any $\infty$-topos in HTT 5.5.6.18, and the fact that slices of $\infty$-topoi are $\infty$-topoi. In other words the unit of the adjunction $\tau_{\leq k}: \mathscr{C}_{/Y} \rightleftarrows \tau_{\leq k}(\mathscr{C}_{/Y}): i$ guarantees a factorization $f = \tau_{\leq k}(f) \circ \eta_f$ such that $\tau_{\leq k}(f)$ is $k$-truncated. However it does not prove that $\eta_f$ is $k$-connected. Rezk proves this in the case that $Y \simeq *$ is the terminal object here. My question then is how proving this special case implies it in the general case.

Answer 1

I think the point is that a map $f\colon A\to B$ in a slice $\mathscr{C}_{/Y}$ is $k$-connected if and only if the underlying map in $\mathscr{C}$ is $k$-connected, and this is supposed to be a formal consequence of the fact that the class of $k$-truncated maps is closed under base change, and the fact that $k$-connected maps are defined to be left orthogonal to $k$-truncated maps. So the construction which factors $A\to Y$ as $A\to E\to Y$, where the maps are $k$-connected/$k$-truncated in $\mathscr{C}_{/Y}$, also produces a factorization into $k$-connected/$k$-truncated in $\mathscr{C}$.