Let $L: \mathcal C^\to_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, $L$ is left exact iff the class of $L$-local morphisms is stable under base change [1].

This appears to be true $\infty$-categorically, as well. Is there a proof in the literature?

~~ But $\infty$-categorically, this is no longer true:~~

**Example:** Let $\mathcal C$ be the $\infty$-category of spaces and $L\mathcal C$ the full subcategory of $n$-truncated spaces for some fixed $n$, so that $L$ is the $n$-truncation functor and the $L$-local morphisms are those with $(n+1)$-connected fibers. Then $L$ is not left exact (failing to preserve, for example, the pullback square $K(\mathbb Z, n) \rightrightarrows \ast, \ast \rightrightarrows K(\mathbb Z, n+1)$), but the $L$-local morphisms are stable under base change.

**Question:** Let $L: \mathcal C^\to_\leftarrow L\mathcal C: i$ be an adjunction of finitely-complete $\infty$-categories with $i$ fully faithful. Let $\mathcal W = L^{-1}(\{\textrm{isos}\}) \subseteq \textrm{Mor} \mathcal C$ be the class of $L$-local morphisms. What are necessary and sufficient closure conditions on $\mathcal W$ ensuring that $L$ is left exact?

~~I'm happy to assume that $\mathcal C$ is presentable, or even an $\infty$-topos, and that $\mathcal W$ is of small generation. ~~

[1] Here we assume that $\mathcal C$ is finitely complete. A morphism $f$ is said to be *$L$-local* if $L(f)$ is an isomorphism, and a class of morphisms $\mathcal W$ is *stable under base change* if $f \in \mathcal W$ implies $f' \in \mathcal W$ where $f'$ is any pullback of $f$ along an arbitrary morphism.

doeshold $\infty$-categorically. I've just written up a proof over here. $\endgroup$ – Tim Campion Jan 8 at 21:30