JOURNAL OF ALGEBRA 8, 96-129 (1967) Representable Functors with Values in Arbitrary Categories* FRIEDRICH ULMER Forschungsimtitut fiir Mathematik der E.T.H., Ziirich, Switzerland and Mathematisches Institut der Universitiit Heidelberg, Heidelberg, Germany Communicated by Saundtrs MacLane Received October 10, 1966; revised December 11, 1966 The representable functors from a category A to the category of sets S have the f&owing basic properties: (1) Let [A, -1 : A-S, AEA, b e a representable functor and t : A -+ S be an arbitrary functor. Then there exists a bijection [[A, -I, t] z tA which is natural in A and t. (2) Each functor t : A +S is canonically a direct limit of representable functors. (In general the index category is not small.) In other words, the Yoneda embedding Y* : A”pp + (A, S), A + [A, -I, is dense (cf. [25] 1.3 and [25] 1.10). In this paper we shall define a concept of “representable functor” in an arbitrary functor category (A, B) in such a way that properties similar to (1) and (2) hold. For this purpose we first consider the case where B is right complete and has a small dense ([2.5] 1.3) subcategory B. Let I : B -+ B denote the inclusion. Then by [25] 1.15 B is a left retract of (%p, S), i.e., the canonical embeddingS : B -+ (BgP, S),B ,- [I--, B],hasaleftadjoint T : (%‘p, S) + B and the end adjunction TS + idB is an equivalence. Therefore the induced functor (A, T) : (A, (B “Pp, S)) + (A, B) is also a left retraction. Let B be an object of B. Denote by B@ : S -+B the functor M-+&EM&, where B, = B. One readily verifies that B@ is left adjoint to [1B, -1 : B-S. * Part of this work was supported by: Fonds fiir akademische Nachwuchsfdrdenmg des Kantons Ziirich. 96