A functor between locally presentable categories is a right adjoint iff it is accessible and preserves limits [1]. We prove that for a number of categories, called graduated, the preservation of countable limits is enough. Examples of graduated categories include sets, posets, Boolean algebras and vector spaces. For certain categories the condition of preserving countable limits may be weakened. Namely, for endofunctors over vector spaces or pointed sets the preservation of (finite) products imply the preservation of (finite) limits. For sets, this is also true except for a certain functor. The talk is based on joint work with J. Adámek presented in [2].

[1] J. Adámek, J. Rosicky, Locally presentable and accessible categories, Cambridge Univ. Press, Cambridge, 1994.

[2] J. Adámek, L. Sousa, A Finitary Adjoint Theorem, arXiv: 2311.14965.