A set is an algebraic structure in the trivial way: one sort, no operations. The theory of sets is an algebraic theory in the trivial way: no equations.

]]>1. What is exactly the definition of homomorphism in this context?

In the definition of homomorphism, nLab says: "More generally, a homomorphism between sets equipped with any algebraic structure is a map preserving this structure."

2. Then All the sets have an algebraic structure, or a set is an algebraic structure itself? ]]>