I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-etale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor $f^{-1}$ sends $N \in X_{lis-et}$ to the sheaf associated to the presheaf which to any $V \in Lis-Et(Y)$ associates the colimit $\lim_{V \rightarrow U} N(U)$ where the colimit is taken over all morphisms over $f$ from $V$ to objects $U \in Lis_Et(X)$. Olsson says this functor $f^{-1}$ is not left exact, because the colimit is not filtering (it is connected but equalizers do not exists). And therefore $(f^{-1}, f_*)$ does not define a morphism of topoi $Y_{lis-et} \to X_{lis-et}$.

I had a couple stupid questions:

1. When $f$ is a map of schemes (with the Zariski topology), I think $f^{-1}$ is exact, since $f^{-1}N_{y} = N_{f(y)}$ correct me if I'm mistaken. My argument roughly is that given any open neighborhood $U$ of $f(y)$, $f^{-1}(U)$ is an open neighborhood of $y$. And so f maps a basis of nghds of y to subsets contained in a basis around $f(y)$, giving the result. Why doesn't the analogous argument hold for the lisse-et topology? I suspect the answer is in whatever it means that "the colimit is not filtering" but I don't know what this means - I see from an example Olsson gives in (3.3) (omitted here) that equalizers need not exists in $Lis-Et(Y)$, but I'm not sure what this has to do direct limits.

1.5. In response to the comments/answers received, here is a more concrete version of what I'm trying to ask in question 1 above - presumably example 3.4 (described below)in Olsson's paper mentioned above is crucially using the lisse-etale topology but does apply to the Zariski topology and I am wondering why - Let $k$ be a field and $X=Spec K(t)$ and $Y=Spec k$ and $f:Y \to X$ the inclusion of the origin. Let $g:{\mathcal{O}}{X_{lisse-et}} \to {\mathcal{O}}{X_{lisse-et}}$ (I'm having trouble getting the $X$ to appear in the subscript) be given by multiplication by $t$. This map has kernel zero. But $f^{-1}\mathcal{O}_X = \mathcal{O}_Y$

as lisse-etale sheaves (because $Hom(f^{-1}\mathcal{O}, G}=Hom(\mathcal{O}, f_*G)=f_*G(\mathbb{A}^1_X)=G(\mathbb{A}^1_Y)$ and $f^{-1}g$ has non-zero kernel. Why doesn't this argument work in the Zariski topology? My guess is that $f^{-1}\mathcal{O}_X = \mathcal{O}_Y$ is not true in the Zariski topology (in fact I calculate $f^{-1}\mathcal{O}_X=k[t]_{(t)}$ but I don't see why the calculation given above doesn't apply.

2. Why does $f^{-1}$ being not left exact imply we don't have a morphism of topoi? For a morphism of topoi, we need $f^{-1}$ to be left adjoint to $f_*$, and hence $f^{-1}$ needs to be right exact.