Definition of diffeological space In this notes A first idea of quantum field theory  the author define a
diffeological space $X$ as

*

*a $\underline{\text { set }} X_{s} \in \underline{\text { Set }} ;$

*for each $n \in \mathbb{N}$ a choice of subset
$$
X\left(\mathbb{R}^{n}\right) \subset \operatorname{Hom}_{\mathrm{Set}}\left(\mathbb{R}_{s}^{n}, X_{s}\right)=\left\{\mathbb{R}_{s}^{n} \rightarrow X_{s}\right\}
$$
of the set of functions from the underlying set $\mathbb{R}_{s}^{n}$ of $\mathbb{R}^{n}$ to $X_{s^{\prime}}$ to be called the smooth functions or plots from $\mathbb{R}^{n}$ to $X$;

*for each $\underline{\text { smooth function }} f: \mathbb{R}^{n_{1}} \rightarrow \mathbb{R}^{n_{2}}$ between Cartesian spaces (def. $1.1$ ) a choice of function
$$
f^{*}: X\left(\mathbb{R}^{n_{2}}\right) \longrightarrow X\left(\mathbb{R}^{n_{1}}\right)
$$
to be thought of as the precomposition operation
$$
\left(\mathbb{R}^{n_{2}} \stackrel{\Phi}{\longrightarrow} X\right) \stackrel{f^{*}}{\mapsto}\left(\begin{array}{cc}
f & \Phi \\
\mathbb{R}^{n_{1}} \rightarrow \mathbb{R}^{n_{2}} \rightarrow X & \rightarrow
\end{array}\right)
$$

*(gluing)
If $\left\{U_{i} \stackrel{f_{i}}{\hookrightarrow} \mathbb{R}^{n}\right\}_{i \in I}$ is a differentiably_good open cover of a Cartesian space (def. 1.5) then the function which restricts $\mathbb{R}^{n}$-plots of $X$ to a set of $U_{i}$-plots

$$X\left(\mathbb{R}^{n}\right) \stackrel{\left(\left(f_{i}\right)^{*}\right.}{\hookrightarrow} \prod_{i} \in  X\left(U_{i}\right)$$
is a bijection onto the set of those tuples $\left(\Phi_{i} \in X\left(U_{i}\right)\right)_{i \in I}$ of plots, which are "matching families" in that they agree on intersections
$\left.\phi_{i}\right|_{U_{i} \cap U_{j}}=\left.\phi_{j}\right|_{U_{i} \cap U_{j}}$
I am not understanding  in part 4 of the definition how the bijection  $f^*_i$ is defined
Is this bijection defined by $f^*_i(\psi)=(\phi_1,..\phi_p)$
where $\psi|_{U_{i}} =\phi_{i}$?
 A: 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).
