Are conservative functors stable under pullback?

Question

Let $A, B$ and $C$ be three small categories and let $f: A\to B$ and $g: C\to B$ two functors such that $g$ is conservative. We can take the pullback as in the following picture: $\require{AMScd}$ \begin{CD} P @>{g^{*}}>> A\\ @VVV @VVV\\ C @>{g}>> B. \end{CD}

My question is: $g^{*}$ is always conservative? In case the answer is no, for which $f$ is $g^{*}$ conservative?

Furthermore, we can canonically rephrase the above situation in the context of $\infty$-category. So, let $A,B$ and $C$ be three small $\infty$-categories and so on ...

My questions become: the $\infty$-functor $g^{*}$ is always conservative? In case the answer is no, for which $\infty$-functor $f$ is $g^{*}$ conservative?

Answer 1

I discovered that the answer is yes because the class of conservative functor is the right class of a factorization system in $Cat_{\infty}$, see Example 3.1.7 (f) in the article "Left-exact Localizations of $\infty$-Topoi I" (Url: file:///C:/Users/teoxd/Desktop/Left-exact%20Localizations%20of%20infty-topoi%20I-Higher%20Sheaves-%20Anel-Biedermann-Finster-%20Andr%C3%A9%20Joyal.pdf).