Let $𝑃$ be a presheaf on a category $\mathrm{𝒞︀}$. The category of elements of $𝑃$ is the displayed category over $\mathrm{𝒞︀}$ whose displayed objects over $𝑐$ are the elements $𝑝\in 𝑃𝑐$, and whose displayed morphisms over $𝑓:𝑐\to 𝑑$ from $𝑝$ to $𝑞$ are proofs that $(𝑃𝑓)(𝑞)=𝑝$.

Since $𝑃𝑐$ is a set, there is at most one displayed morphism over each $𝑓$ between given elements: a morphism of elements is a morphism of $\mathrm{𝒞︀}$ that happens to carry $𝑞$ back to $𝑝$. Its total category is the classical category of elements $\int 𝑃$, and a universal element of $𝑃$ is exactly a terminal object of $\int 𝑃$.