For a scheme $X$, let $LE(X)$ denote the lisse-etale site on $X$. This is the full subcategory of $\textbf{Sch}/X$ consisting of smooth morphisms to $X$, equipped with the etale topology. Let $\mathcal{O}_X$ denote the presheaf on $LE(X)$ sending $$(U\rightarrow X)\mapsto \Gamma(U,\mathcal{O}_U)$$ This is in fact represented by $\mathbb{A}^1_X$, and hence is a sheaf (of sets)

Now let $k$ be an alg. closed field. Let $X := \mathbb{A}^1_k$ with coordinate $t$. Let $pt := \text{Spec }k$.

Consider the map $i : pt\rightarrow X$ sending $pt\mapsto (t=0)$. By Yoneda, it is possible to show that $i^*\mathcal{O}_X = \mathcal{O}_{pt}$ (ie, the inverse image sheaf of the functor of points of $\mathbb{A}^1_X$ in $LE(X)$ is the functor of points of $\mathbb{A}^1_{pt}$ in $LE(pt)$).

In particular, $i^*\mathcal{O}_X(pt) = k$. **I don't see how this agrees with the definition of $i^*$.**

By definition, $i^*\mathcal{O}_X$ is the sheafification of the presheaf defined as follows:

For every $Z\in LE(pt)$, consider the category $I_Z$ consisting of objects $(U,\rho)$ where $U\in LE(X)$ and $\rho : Z\rightarrow U_{pt} := U\times_X pt$ is a morphism in $LE(pt)$. A morphism $(U,\rho)\rightarrow (U',\rho')$ is a morphism $g : U\rightarrow U'$ in $LE(X)$ such that $g_{pt}\circ \rho = \rho'$.

Then, we consider the presheaf on $LE(pt)$: $$(i^*)^{pre}\mathcal{O}_X : (Z\rightarrow pt)\mapsto \varinjlim_{(U,\rho)\in I_Z^{op}} \mathcal{O}_X(U) = \varinjlim_{(U,\rho)\in I_Z^{op}}\Gamma(U,\mathcal{O}_U)$$ where given $h : Y\rightarrow Z$ in $LE(pt)$, we get a restriction map given by the functor $I_Z\hookrightarrow I_Y$ sending $(U,\rho)\mapsto (U,\rho\circ h)$. Then, we define $i^*\mathcal{O}_X$ to be the sheafification of this presheaf.

(So far this is the setup on p45-47 of Olsson's "Algebraic spaces and stacks")

My issue is: by the definition of $i^*\mathcal{O}_X$, there is a natural map $$k[t] = \mathcal{O}_X(X)\rightarrow i^*\mathcal{O}_X(pt) = k$$ Which can't be injective, and for example presumably sends $t,t^2$ both to 0. However, the global sections of the presheaf: $$(i^*)^{pre}\mathcal{O}_X(pt) = \varinjlim_{(U,\rho)\in I_{pt}^{op}}\mathcal{O}_X(U),$$ contains $\mathcal{O}_X(X)$ as a subset (since every $U\rightarrow X$ is smooth, hence flat, hence induces injections on global sections). Thus, the images of $t,t^2$ are distinct in $(i^*)^{pre}\mathcal{O}_X(pt)$. This means that if $t,t^2$ both map to 0 in $i^*\mathcal{O}_X(pt)$, they must agree in some etale covering of $pt$, but this seems impossible since $k$ is alg. closed, so the etale covers carry no additional information.

EDIT: Here is the argument that $i^*\mathcal{O}_X\cong\mathcal{O}_{pt}$. We have, for any sheaf $F$ on $LE(pt)$: $$Hom_{LE(pt)}(i^*h_{\mathbb{A}^1_X},F) = Hom_{LE(X)}(h_{\mathbb{A}^1_X},i_*F) = (i_*F)(\mathbb{A}^1_X) := F(i^*\mathbb{A}^1_X) = F(\mathbb{A}^1_{pt}) = Hom_{LE(pt)}(h_{\mathbb{A}^1_{pt}},F)$$ where we've used Yoneda twice, in the second and final equalities. But then, since this holds for any sheaf $F$ on $LE(pt)$, comparing the last term with the first, by covariant Yoneda we find that $i^*h_{\mathbb{A}^1_X} = h_{\mathbb{A}^1_{pt}}$ Since $\mathcal{O}_X$ is represented by $\mathbb{A}^1_X$ and $\mathcal{O}_{pt}$ is represented by $\mathbb{A}^1_{pt}$, this shows that $i^*\mathcal{O}_X \cong \mathcal{O}_{pt}$.