Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ canonically isomorphic to the cocharacter lattice $L$ of $T$?

If $\operatorname{Spec}A$ is affine scheme, it seems to me that $p^*T=T\times \mathbb{G}_m$, so $$p_*p^*T(A)=p_*((T\times \mathbb{G}_m))(A)=T(A[t^{\pm 1}])\times \mathbb{G}_m(A[t^{\pm 1}]).$$ Say that $A=K$ is a field. Then $$p_*p^*T(K)=L\oplus\mathbb{Z},$$ but why is the image of $T(K)$ equal to $\mathbb{Z}$?