Definition of diffeological space

Solution 1:

According to this definition, a differentiably good open cover $\{U_i → \mathbb{R}^n\}$ is a family of smooth maps $f_i:\mathbb{R}^{n_i} → \mathbb{R}^n$ between Cartesian spaces (satisfying some properties). In particular, one can apply $(-)^*$ to each of these maps, yielding maps $f_i^*:X(\mathbb{R}^n) → X(\mathbb{R}^{n_i})$. These induce the desired map $(f_i^*)_{i ∈ I}:X(\mathbb{R}^n) → ∏_iX(\mathbb{R}^{n_i})$ which sends $ψ$ to the tuple $(f_i^*(ψ))_{i ∈ I}$.

Thinking of $f_i^*$ as a precomposition operation one could denote this tuple by $(ψ ∘ f_i)_{i ∈ I}$. But be aware that this is only notation, it is not actually possible to compose $ψ$ and $f_i$ since $ψ$ is not a function but an element of $X(\mathbb{R}^n)$.

Furthermore, since each $f_i$ is required to be an open embedding, we can identify its domain with its image in $\mathbb{R}^n$ (as sets), which is denoted by $U_i$. Surpressing the inclusion map, the tuple $(ψ ∘ f_i)_{i ∈ I}$ could then also be denoted by $(ψ\mid_{U_i})_{i ∈ I}$ making your intuition about the map $(f_i^*)_{i ∈ I}$ quite right (just note that this tuple might have infinitely many entries, not just finitely many as in your question).