Identifiability is constancy on observational classes

Equality of measures is reflexive, symmetric and transitive; by [F2], $\sim$ is therefore an equivalence relation, so each class $[\theta {]}_{\sim }$ is defined.

If $\psi =\bar{\psi }\circ \mathsf{P}$, then $P_{\theta }=P_{\eta }$ gives $\bar{\psi }(P_{\theta })=\bar{\psi }(P_{\eta })$ and hence $\psi (\theta )=\psi (\eta )$. Thus (iii) implies identifiability.

Suppose $\psi$ is identifiable. Form the relation $G:=\{(Q,a):\text{there exists }\theta \in \mathrm{\Theta }\text{ with }Q=P_{\theta }\text{ and }a=\psi (\theta )\}$. Every $Q\in \mathcal{M}$ has at least one such parameter by [F1]. If both $\theta$ and $\eta$ represent the same $Q$, then [F2] gives $\theta \sim \eta$, so [F4] gives $\psi (\theta )=\psi (\eta )$. Thus $G$ has exactly one value for each $Q\in \mathcal{M}$, defines a function $\bar{\psi }:\mathcal{M}\to {\mathcal{T}}_0$ by [F3], and satisfies $\psi =\bar{\psi }\circ \mathsf{P}$. This construction defines the graph directly and uses no choice axiom.

By [F2], $\eta \in [\theta {]}_{\sim }$ exactly when $\eta \sim \theta$. If $a,b\in [\theta {]}_{\sim }$, then $a\sim \theta$ and $b\sim \theta$, so symmetry and transitivity from step 1.1 give $a\sim b$; conversely, $\theta \sim \eta$ gives $\eta \in [\theta {]}_{\sim }$. Hence [F4] is exactly constancy of $\psi$ on every class.

If ${\bar{\psi }}_1$ and ${\bar{\psi }}_2$ both factor $\psi$, each $Q\in \mathcal{M}$ equals $P_{\theta }$ for some $\theta$ by [F1], and then ${\bar{\psi }}_1(Q)=\psi (\theta )={\bar{\psi }}_2(Q)$ by [F3]. The factor is unique. The implication in step 1.2 and the construction in step 1.3 give (i) iff (iii), while step 2.1 gives (i) iff (ii).