Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. Hylomorphisms form a profunctor from coalgebras to algebras,

where $\text{𝖧𝗒𝗅𝗈}((𝑋,𝛾),(𝐵,𝛼))$ is the set of $ℎ:𝑋\to 𝐵$ with $ℎ=𝛾\star 𝐹ℎ\star 𝛼$.

The action is by composition. If $𝑔:(𝑌,𝛿)\to (𝑋,𝛾)$ is a coalgebra morphism ($𝑔\star 𝛾=𝛿\star 𝐹𝑔$) and $𝑘:(𝐵,𝛼)\to (𝐵\prime ,𝛼\prime )$ is an algebra morphism ($𝛼\star 𝑘=𝐹𝑘\star 𝛼\prime$), then $𝑔\star ℎ\star 𝑘$ is again a hylomorphism:

In these terms, a coalgebra $𝛾$ is recursive when $\text{𝖧𝗒𝗅𝗈}(𝛾,-)$ is the terminal functor, and an algebra $𝛼$ is corecursive when $\text{𝖧𝗒𝗅𝗈}(-,𝛼)$ is. For the inverse of an initial algebra the profunctor is representable, $\text{𝖧𝗒𝗅𝗈}({\text{𝗂𝗇}}^{-1},-)\cong \text{𝖠𝗅𝗀}(𝐹)(\text{𝗂𝗇},-)$, and dually $\text{𝖧𝗒𝗅𝗈}(-,{\text{𝗈𝗎𝗍}}^{-1})\cong \text{𝖢𝗈𝖺𝗅𝗀}(𝐹)(-,\text{𝗈𝗎𝗍})$ for a terminal coalgebra.

When every value of $\text{𝖧𝗒𝗅𝗈}$ is a singleton, every divide-and-conquer specification over $𝐹$ has exactly one solution. This is what local contractivity guarantees.