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Adjoint Functor admits a codensity algebras base functor category theory class of V-natural closed category codensity V-monad codomain coequalizer cogenerating colimits comma category commutativity of diagram complete corealization correspondence cotensors defined definition denote dual end of V-functors enriched categories equal equation equivalent by rQ fact ffi(A ffi(V following diagram commutative formula functor categories Given a V-functor hence Homology Theory Jc T(C Lemma level of sets limit maps monad monomorphism morphism naturally in F ordinal preserves Proof Proposition III.1.1 Rang(id RanG(id)G Rang(S Rans(R representable functors right Kan extensions right Yoneda semantical comparison V-functor small V-category small V-limits tensors tion tractable universal property V-adjoint V-cocomplete V-cocontinuous V-codense V-coequalizers V-cogenerating V-complete V-category V-context V-continuous small V-continuous V-dense V-faithful V-full-and-faithful V-functor V-functor CB V-left adjoint V-lim r(X V-limit of V-functors V-monomorphism V-natural families V-natural isomorphisms V-natural transformations V-structure ZA(A