Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. A terminal $𝐹$-coalgebra, written $𝜈𝐹$, is a terminal object of the category of coalgebras $\text{Coalg}(𝐹)$ — equivalently, an initial algebra for ${𝐹}^{\text{op}}$.

Unfolding the universal property: a terminal coalgebra is a coalgebra $(𝜈𝐹,\text{out})$ such that every coalgebra $(𝑥,𝛾)$ admits a unique morphism $\text{unfold}𝛾:𝑥\to 𝜈𝐹$ satisfying