§4.1, eq. (13), p. 34 — the local construction F(X, A)ᵤ = φ(xᵤ, X_{Nᵤ}) satisfies (10)

Let $\mathbf A\in\mathbb R^{n\times n}$ be an adjacency matrix and $\mathbf X\in\mathbb R^{n\times d}$ node features. For a node $u$ let $\mathcal N_u=\{v:a_{uv}\ne0\text{ or }a_{vu}\ne0\}$ be its neighbourhood (11) and $\mathbf X_{\mathcal N_u}=\{\!\{\mathbf x_v:v\in\mathcal N_u\}\!\}$ the multiset of its neighbours' features (12). Let $\phi:\mathbb R^d\times\{\text{finite multisets of }\mathbb R^d\}\to\mathbb R^{d'}$ be any function, and let $\mathbf F(\mathbf X,\mathbf A)$ be the matrix whose $u$-th row is $\phi(\mathbf x_u,\mathbf X_{\mathcal N_u})$, eq. (13). Then $\mathbf F$ is permutation equivariant in the sense of (10):

This is the book's general recipe for permutation-equivariant functions on graphs: a shared local function that is invariant to the order of the neighbours.

Formalization Note Since $\phi$ receives a multiset, its output does not depend on the ordering of the nodes in $\mathcal N_u$ — the book's hypothesis "$\phi$ is permutation invariant" holds by construction. Edges are the nonzero entries of $\mathbf A$, which for the 0/1 matrix (8) is the book's edge set.