It is known a version of Adjoint Functor Theorem for locally presentable categories, which says that a functor between such categories has a left adjoint iff it is continous (i.e. preserves all limits) and accessible (preserves $\lambda$-filtered colimits for some cardinal $\lambda$) (see for this Theorem 1.66 of J. Adamek, J. Rosicky, "Locally Presentable and Accessible Categories", London Math. Soc. Lecture Notes, Cambridge, 1994). My question is then to find an example of a continous functor between locally presentable categories which is not accessible.



Example of a continous functor between locally presentable categories which has no left adjoint
The following example was coincidentally mentioned by André Joyal on the categories mailing list today; he attributed it to Mac Lane. For every infinite cardinal number k, let $G\_k$ be a simple group of cardinality k. Define the functor ML: Group → Set to be the product of all the representable functors $\mathrm{Hom}(G\_k,-)$. Since no group can admit a nontrivial homomorphism from proper-class-many of the $G\_k$, this functor does indeed land (or can be redefined to land) in Set. Since it is a product of representables, it is continuous (and of course Group and Set are locally presentable), but it is not itself representable (hence has no left adjoint).


