Recall the notion of a final functor, which is a sort of colimit-preservation property. Is such class of functors stable under pullbacks in Cat? Namely, is the pullback of a final functor along any other functor still final? If not, what is a counterexample?
A reference would also be welcome.
No, final functors are not stable under pullback in general.
Let $I := \{0\to1\}$ be the walking arrow category, then $1:*\to I$ picks out the terminal object and is thus final. The diagram $\require{AMScd}$ \begin{CD} \varnothing @>>> *\\ @V{F}VV @VV{1}V\\ * @>>{0}> I \end{CD} is a pullback square, but the functor $F:\varnothing\to*$ is not final. Indeed, let $G:*\to\mathbf{Set}$ pick any nonempty set $X$, then $\varinjlim G=X$, but $\varinjlim G\circ F=\varnothing$ since the colimit of the empty diagram is just the initial object in $\mathbf{Set}$. As the unique map $\varnothing\to X$ is not bijective, we can conclude that $F$ is not final despite the finality of $1:*\to I$.
However, final functors form an orthogonal factorisation system with discrete fibrations, and so in particular are closed under pushouts in $\mathbf{Cat}$ (see e.g., here)