Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. Coalgebras require no new construction: a coalgebra is an algebra in the opposite category. Define

a displayed category over ${\mathrm{𝒞︀}}^{\text{op}}$, where ${𝐹}^{\text{op}}:{\mathrm{𝒞︀}}^{\text{op}}\to {\mathrm{𝒞︀}}^{\text{op}}$ is $𝐹$ acting on the opposite category.

Concretely, over an object $𝑥$ a displayed object is a structure map

The category of coalgebras is the opposite of the total category:

The outer opposite returns morphisms to the direction of $\mathrm{𝒞︀}$: a morphism $(𝑥,𝛾)\to (𝑦,𝛿)$ is a map $𝑓:𝑥\to 𝑦$ with