Any two natural endomorphisms $\alpha,\beta$ of an identity functor commute. This fact follows from observing the equation $\alpha_A\circ \beta_A=\beta_A\circ\alpha_A$ is a naturality square for either $\alpha$ or $\beta$.
Thus the endomorphism monoid of an identity functor is commutative. Consequently one-sided inverses are automatically two-sided, and this fact can be used e.g to prove that if $F\dashv G$ admits an isomorphism $1\cong GF$ then the unit of adjunction is also an isomorphism.
Question. What other kinds of functors have commutative endomorphism monoids?