This is the transduction-based definition of monadic dependence. A general transduction also has a domain formula, which selects the vertices of the output graph. It is not needed here, because the vertex set of H is the image of f, and the class of all graphs is closed under induced subgraphs. A transduction with domain formula δ(x) and edge formula ψ that produces every graph gives an f as above (map H to the vertices of its copy that δ selects). Conversely, f gives the transduction whose domain formula is a fresh color marking the image of f and whose edge formula is ψ(x₀, x₁) ∧ x₀ ≠ x₁. The conjunct x₀ ≠ x₁ is needed because the condition above constrains ψ on distinct vertices only, while the edge formula of a transduction defines the adjacency on all pairs of selected vertices, and a vertex is not adjacent to itself.

The definition of the paper is of this form. It starts from a formula φ(x, y), takes the graph with the edges uv such that φ(u, v) or φ(v, u) holds, and asks for every finite graph as an induced subgraph. Such a φ gives ψ = φ(x₀, x₁) ∨ φ(x₁, x₀) here. Conversely, a ψ as above serves as φ, since on the image of f it holds of u and v in both orders or in neither. So the two definitions agree.

Monadic dependence is often defined by forbidding the independence property for formulas on the colorings of the graphs in C. By a theorem of Baldwin and Shelah the two definitions agree.