The (possibly infinitary) diagram formula $\Phi_{\mathcal{A}}(X,Y)$ is then:

$\exists V[\bigwedge_{a,b \in A; a \neq b} (x_a \neq x_b) \wedge \bigwedge_{a \in A} Xx_a \wedge \forall x(Xx \to \bigvee_{a \in A} (x = x_a))$ $\wedge \bigwedge_{a,b \in A} (\pm_{ab} Yx_a x_b) ]$where $V = \{x_a \mid a \in A\}$ and $\pm_{ab}Yx_ax_b$ is $Yx_ax_b$ if $(a,b) \in R$ and $\neg Yx_ax_b$ otherwise.

On the Diagram Conception of Abstract Structure,

TheCategoricity ensures that, for any $\mathcal{B} = (B,S)$ (a relational model, with a single binary relation $S \subseteq B^2$), we have:abstract structureof $\mathcal{A}$ is theproposition$\hat{\Phi}_{\mathcal{A}}$ expressed by the formula $\Phi_{\mathcal{A}}(X,Y)$.

$\mathcal{B} \models \Phi_{\mathcal{A}}(X,Y)$ if and only if $\mathcal{B} \cong \mathcal{A}$.

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