Associated to a simple root of a finite-dimensional complex semisimple Lie algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel, Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category $\mathcal{O}$. We study their relations, compute cohomologies of their derived functors and describe the monoid generated by Arkhipov's and Joseph's functors and the monoid generated by Irving's functors. Natural transformations between elements of these monoids are investigated. It turns out that the endomorphism rings of all elements in these monoids are isomorphic. We also use Arkhipov's, Joseph's and Irving's functors to produce new generalized tilting modules.