Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite continuous $G$-sets with discrete topology are (denoted by $G-Set_f$) anti-equivalent.

However, it is well known that the category $G-Set_f$ is a (elementary) topos, and so is $Sep(K)^{op}$. Especially, it is cartesian closed. I think the product of this category (the coproduct of $Sep(K)$) is just a tensor product $A\otimes_K B$. So, the exponentials $C^B$ have to satisfy $Hom_{Sep(K)^{op}}(A\otimes _K B, C)\simeq Hom_{Sep(K)^{op}}(A,C^B)$ (equivalently $Hom_{Sep(K)}(C,A\otimes _K B)\simeq Hom_{Sep(K)}(C^B,A)$) , but I don't know such objects.

So my question is the followings.

• What is the exponential object of $Sep(K)^{op}$?
• What is the subobject classifier of $Sep(K)^{op}$?

More generally, let $X$ be a connected scheme and $\pi(X,x)$ be its fundamental group for some fixed geometric point $x$. Then, the category of finite etale algebras over $X$ (denoted by $FEt(X)$) and $\pi(X,x)-Set_{f}$ are anti-equivalent. In this case, what is its exponentials?