A universal element of a presheaf $𝑃$ on a category $𝐶$ is an element $𝑥\in 𝑃𝑐$, where $𝑐$ is some object of $𝐶$, demonstrating that $𝑃$ is representable by $𝑐$.

(Slightly) more concretely, a universal element is captured by the following three pieces of data

• An object $𝑐$ of $𝐶$
• An element $𝑥\in 𝑃𝑐$
• A proof that the map sending a morphism $𝑓:𝑏\to 𝑐$ to $(𝑃𝑓)(𝑥)$ is an equivalence

This third point states that morphisms from $𝑏$ into $𝑐$ are uniquely determined by an element of $𝑃$ at the domain $𝑏$.

Or equivalently, universal elements are terminal in the category of elements.