Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. An initial $𝐹$-algebra, written $𝜇𝐹$, is an initial object of the category of algebras $\text{Alg}(𝐹)$.

Unfolding the universal property: an initial algebra is an algebra $(𝜇𝐹,\text{in})$ such that every algebra $(𝑥,𝛼)$ admits a unique morphism $\text{fold}𝛼:𝜇𝐹\to 𝑥$ satisfying