The Abstract Structure of a Binary Relational Model

Generalizing the previous M-Phi post on the abstract structure of an $\omega$-sequence, suppose that $\mathcal{A}=(A,R)$ is any (set-sized) binary relational model (i.e., $A$ is the domain/carrier set and $R\subseteq A^2$). Let $\kappa =|A|$. Let $L$ be a second-order, possibly infinitary, language, with $=$ (but no non-logical primitive symbols), which allows compounds over $\kappa$-many formulas and allows quantifier prefixes to be a set $V$ of variables of cardinality $\kappa$. For each $a\in A$, let $x_a$ be a unique variable that "labels" $a$. Let the second-order unary variable $X$ label the domain $A$ and let the second-order binary variable $Y$ label the relation $R$.

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

$\mathrm{\exists }V[\underset{a,b\in A;a\ne b}{\bigwedge }(x_a\ne x_b)\wedge \underset{a\in A}{\bigwedge }X{x}_a\wedge \mathrm{\forall }x(Xx\to \underset{a\in A}{\bigvee }(x=x_a))$ $\wedge \underset{a,b\in A}{\bigwedge }({\pm }_{ab}Y{x}_a{x}_b)]$

where $V=\{x_a\mid a\in A\}$ and ${\pm }_{ab}Y{x}_a{x}_b$ is $Y{x}_a{x}_b$ if $(a,b)\in R$ and $\mathrm{\neg }Y{x}_a{x}_b$ otherwise.

On the Diagram Conception of Abstract Structure,

The abstract structure of $\mathcal{A}$ is the proposition ${\hat{\mathrm{\Phi }}}_{\mathcal{A}}$ expressed by the formula ${\mathrm{\Phi }}_{\mathcal{A}}(X,Y)$. 

Categoricity ensures that, for any $\mathcal{B}=(B,S)$ (a relational model, with a single binary relation $S\subseteq B^2$), we have:

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