Homomorphism commutes with word maps

Statement
 Suppose $$w$$ is a word in the letters $$x_1,x_2,\dots,x_n$$ (these are just formal symbols). Suppose $$\varphi:G \to H$$ is a homomorphism of groups. Then, $$\varphi$$ commutes with $$w$$, i.e.:

$$\varphi(w(g_1,g_2,\dots,g_n)) = w(\varphi(g_1),\varphi(g_2),\dots,\varphi(g_n)) \ \forall \ g_1,g_2,\dots,g_n \in G$$

where the $$w$$ on the left is the word map in $$G$$ (i.e., it evaluates the word for a tuple of values of the letters in $$G$$ and the $$w$$ on the right is the word map in $$H$$.

Applications

 * Verbal implies fully invariant

Proof
A formal proof can be given by inducting on the length of the word.