Generally, a homomorphism between two algebraic objects $A,B$ is a function $f:A→B$ which preserves the algebraic structure on $A$ and $B$. That is, if elements in $A$ satisfy some algebraic equation involving addition or multiplication, their images in $B$ satisfy the same algebraic equation. We note that both $A$ and $B$ are in the same variety.
My question is about defining a homomorphism between two algebraic objects from different varieties. Is it possible to define such function? If so, how?
Without really knowing what you have in mind and for what purpose, a reasonable general idea is the following:
You have three types of objects, $U,V,W$. You have a way $F$ to transform an object of type $U$ to an object of type $W$, and you have a way $G$ to transform an object of type $V$ to an object of type $W$.
Then given objects $A$ and $B$ of type $U$ and $V$ respectively, define a generalized homomorphism from $A$ to $B$ to be a $W$-homomorphism between $F(A)$ and $G(B)$.
In terms of category theory, $U,V,W$ should be taken to mean categories and $F,G$ to mean functors, and this construction describes the objects of the comma category $(F \downarrow G)$.