synthetic-adjunctions

Adjunctions automate the derivation of universal structures by generating unit and counit data that satisfy triangle identities. This Skill produces adjunction data for limits, colimits, Kan extensions, and monads from directed type theory definitions, enabling synthesis of left and right adjoints with verified coherence.

Unit (η) and counit (ε) are natural transformations that establish the adjoint relationship between functors. This Skill generates both components with automatic verification of triangle identities, ensuring the adjunction is coherent and suitable for constructing universal properties.

Yes. Adjunctions provide the foundational data for synthesizing Kan extensions and monads. This Skill generates the required adjunction structure—unit, counit, and triangle identities—from free-forgetful pairs and other universal properties, enabling downstream monad and Kan extension construction.

Triangle identities ensure unit and counit compose correctly. This Skill automatically generates and validates both identities as part of adjunction synthesis, confirming coherence without manual proof.

Provide a functor definition, universal property characterization, or free-forgetful pair structure. The Skill synthesizes the adjoint functor, unit, counit, and triangle identities from these specifications, with support for Agda and Haskell-like code representations.