Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. An algebra $𝛼:𝐹𝐵\to 𝐵$ is corecursive when for every coalgebra $𝛾:𝑋\to 𝐹𝑋$ there is exactly one hylomorphism from $𝛾$ to $𝛼$. Equivalently, the functor $\text{𝖧𝗒𝗅𝗈}(-,𝛼)$ of the hylomorphism profunctor is constantly a singleton. It is the dual of a recursive coalgebra: an $𝐹$-algebra in $\mathrm{𝒞︀}$ is corecursive exactly when it is recursive as an ${𝐹}^{\text{op}}$-coalgebra in ${\mathrm{𝒞︀}}^{\text{op}}$.

Example. If $(𝜈𝐹,\text{𝗈𝗎𝗍})$ is a terminal coalgebra, then $\text{𝗈𝗎𝗍}$ is invertible and ${\text{𝗈𝗎𝗍}}^{-1}:𝐹(𝜈𝐹)\to 𝜈𝐹$ is a corecursive algebra: a solution of $ℎ=𝛾\star 𝐹ℎ\star {\text{𝗈𝗎𝗍}}^{-1}$ is the same thing as a solution of $ℎ\star \text{𝗈𝗎𝗍}=𝛾\star 𝐹ℎ$, that is, a coalgebra map into the terminal coalgebra, and there is exactly one, $\text{𝗎𝗇𝖿𝗈𝗅𝖽}𝛾$.