universal relation

If $\mathrm{\Phi }$ is a class of $n$-ary relations with $\overrightarrow{x}$ as the only free variables, an $n+1$-ary formula $\psi$ is universal for $\mathrm{\Phi }$ if for any $\varphi \in \mathrm{\Phi }$ there is some $e$ such that $\psi (e,\overrightarrow{x})\leftrightarrow \varphi (\overrightarrow{x})$. In other words, $\psi$ can simulate any element of $\mathrm{\Phi }$.

Similarly, if $\mathrm{\Phi }$ is a class of function of $\overrightarrow{x}$, a formula $\psi$ is universal for $\mathrm{\Phi }$ if for any $\varphi \in \mathrm{\Phi }$ there is some $e$ such that $\psi (e,\overrightarrow{x})=\varphi (\overrightarrow{x})$.