On the smallest and the largest elements of the Lockett section of a fitting functor

In the paper analogues of the known in the theory of Fitting classes operators «$^*$», «$_ *$ » and the Lockett sections on the set of Fitting $\mathfrak{X}$-functors ($\mathfrak{X}$ is some non-empty Fitting class) are defined. By the Lockett section of a conjugate Fitting $\mathfrak{X}$-functor $f$ we mean the set $\mathrm{Locksec}(f) = \{g: g\text{ is a conjugate Fitting }\mathfrak{X}\text{-functor and }f^* = g^*\}$. It is proved that the Lockett section of a conjugate Fitting $\mathfrak{X}$-functor contains the largest element. Besides we describe conditions under which the Lockett section contains the smallest element.