In the book Mathematical Aspects of Quantum Field Theory (2010) on page 159-160 in chapter 6.6 Wick rotations and axioms for Euclidean QFT the following is stated:
We have seen earlier in this chapter that the Wightman functions $$W(x_1,...,x_n) = ⟨Ω,φ(x_1)···φ(x_n)Ω⟩$$ can be used to completely reconstruct a scalar bosonic field theory. These functions, as it turns out, can be analytically continued to the so-called Schwinger points $z_j = (ix^0_j , x^1_j , x^2_j , x^3_j )$, provided the points $x_j ∈ \mathscr{R}^4$ are such that $x_j\ne x_j$ if $j\ne k$. The map $(\mathbb{R}^4)^n → (i\mathbb{R}×\mathbb{R}^3)^n$ given by $(x_1,x_2 ...,x_n) → (z_1,z_2,...,z_n)$ is called a Wick rotation. The functions $$S_n(x_1,x_2,...,x_n) = W(z_1,z_2,...,z_n)$$ are called Schwinger functions.
(I did not make the typos it is as it is in the book)
But this makes it seems like the Wightman functions are from $(\mathbb{R}^4)^n$ to $\mathbb{C}$ whereas previously in the book they were introduced as the following from page 138
Definition 6.2 The $n$-th Wightman correlation function of a quantum field theory satisfying the Wightman axioms is a function $W_n : \mathscr{S} (\mathbb{R}^4)×\mathscr{S} (\mathbb{R}^4)× ···\mathscr{S}(\mathbb{R}^4) → \mathbb{C}$ given by $$W_n(f_1,f_2,...,f_n) = ⟨ψ_0,φ(f_1)φ(f_2)···φ(f_n)ψ_0⟩,$$ for all $f_j ∈ \mathscr{S} (\mathbb{R}^4)$, where $ψ_0$ is the vacuum vector of the theory and $φ$ is its field operator.
