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

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