  For a new class of algebras, called $EMV$-algebras, every idempotent element
$a$ determines an $MV$-algebra which is important for the structure of the
$EMV$-algebra. Therefore, instead of standard homomorphisms of $EMV$-algebras,
we introduce $EMV$-morphisms as a family of $MV$-homomorphisms from
$MV$-algebras $[0,a]$ into other ones. $EMV$-morphisms enable us to study
categories of $EMV$-algebras where objects are $EMV$-algebras and morphisms are
special classes of $EMV$-morphisms. The category is closed under product. In
addition, we define free $EMV$-algebras on a set $X$ with respect to
$EMV$-morphisms. If $X$ is finite, then the free $MV$-algebra on $X$ is a free
$EMV$-algebras. For an infinite set $X$, the same is true introducing a
so-called weakly free $EMV$-algebra.
