Let $l/k$ be a finite Galois extension of fields of characteristic zero. Let $X$ be an affine scheme of finite type over $l$ and denote the Weil restriction by $\prod_{l/k} X$ (it exists in this case).
Is there a map $\prod_{l/k} X \to X$?
There should be a canonical map $(\prod_{l/k} X) \times_k l \to X$ since in this case $(\prod_{l/k} X) \times_k l \cong \prod_{\sigma \in Gal(l/k)}X^\sigma$ and so we can project the factor $X^\sigma$ for $\sigma$ the identity in $Gal(l/k)$.
The reason for this question is because I saw in a book (which I cannot remember now) that they used this map, but I'm not so sure if this map exists and I think it was just a mistake in the book.
In general such a morphism over $k$ won't exist.
Let's look at an explicit example : $k = \mathbb{R}$, $l = \mathbb{C}$, and $X = \mathbb{G}_m = Spec(A)$ over $\mathbb{C}$, with $A = \mathbb{C}[x,y]/(xy - 1)$.
Then $\prod_{\mathbb{C}/\mathbb{R}}X = Spec(B)$ with $B = \mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$ (I took the example from the accepted answer in https://mathoverflow.net/questions/7715/what-is-restriction-of-scalars-for-a-torus )
Then what you want in a $\mathbb{R}$-algebra morphism $A\to B$. This implies constucting a $\mathbb{R}$-algebra morphism $\mathbb{C}\to B$, ie finding an element $a\in B$ such that $a^2 = -1$. Now $a = \frac{P(x,y)}{(x^2+y^2)^m}$, and we ask $P(x,y)^2 = -(x^2+y^2)^{2m}$, which is impossible just looking at the sign of a term of highest degree.
So there is no such morphism, and thus no $k$-scheme morphism $\prod_{l/k}X \to X$.