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 $i^\*\mathcal{O}\_X \cong \mathcal{O}\_{pt}$.