In this research, we study the structure and properties of the endomorphism monoid of a strong semilattice of left simple semigroups. In such semigroup, we consider mainly that the defining homomorphisms are constant maps or isomorphisms. For arbitrary defining homomorphisms the situation is in general extremely complicated, we have discussed some of the problems at the end. We obtain results for strong semilattices of groups which are known under the name of Clifford semigroups and we also consider strong semilattices of left or right groups as well. Both are special cases of the strong semilattices of left simple semigroups.