Summary: If \(\mathbf{G}=(G;\cdot)\) is a semigroup, \(I\) is arbitrary set and \(\lambda, \rho :I\rightarrow I\) are mappings satisfying the equalities \(\lambda \lambda =\lambda\), \(\rho\rho =\rho\) and \(\lambda\rho =\rho\lambda\) then we define the semigroup \((G^I,\times)\) where \((x\times y)(i) := x(\lambda i)\cdot y(\rho i)\). This construction gives rise to four covariant and two contravariant functors and constitute three adjoint situations. We apply this functors for finding representation theorems.