Is a pre-Lie algebra homomorphism multiplicative? A pre-Lie algebra $A$ is a vector space with a binary operation $(x, y) \mapsto xy$
satisfying
$(xy)z − x(yz) = (yx)z − y(xz)$ for all $x, y, z$.
I assume that a homomorphism of pre-Lie algebras is defined as a linear map that preserves the above relation. Am I right in assuming this definition?
Further, does this imply that the homomorphism is multiplicative? I could use help in showing this.

It does not really mean anything that the map "preserves the relation"; what is preserves is the product.
Consider the corresponding property for a morphism of associative algebras: a morphism is a linear map that preserves the product. Asking it to "preserve the associativity" is not really meaningful.
