Verbality satisfies image condition

Statement
Suppose $$G$$ is a group, $$H$$ is a verbal subgroup of $$G$$, and $$\varphi:G \to K$$ is a homomorphism of groups. Then, the image $$\varphi(H)$$ is a verbal subgroup of $$K$$. In fact, if $$H$$ is the span of a collection of words $$C$$ in terms of the elements of $$G$$, then $$\varphi(H)$$ is the span of the same collection of words in terms of the elements of $$K$$.

Related facts

 * Verbality is transitive
 * Verbality is quotient-transitive
 * Verbality is strongly join-closed