Kerodon

Apologies for the long comment.

I do not believe that this is a straightforward consequence of Remark 7.1.4.8, because changing out $U$ for $V$ actually changes the categories for which we're attempting to prove that something is $U$ (resp $V$) final. That is, we must show that $\bar{f}$ is final for $\mathcal{C}_{/f} \to \mathcal{D}_{/U \circ f}$ iff it's final for $\mathcal{C}_{/f} \to \mathcal{D}_{/V \circ f}$. Unlike in 7.1.4.8, the codomains of these maps are different, so this does not follow merely by pre- and post-composing with a natural isomorphism. It seems easier, in fact, to prove this Remark with the characterization from 7.1.5.8, since we can then cook up equivalences between things sliced over $U \circ f$ and $V \circ f$ and derive the conclusion from this.

Yep. Thanks!