I'm reading the following fragment in the paper "Notes on compact quantum groups":

While I'm familiar with the multiplier algebra (constructed via double centralizers) and its universal property in terms of essential ideals, I'm a little bit unsure why one can extend the map $$\mathcal{B}_0(\mathcal{H}) \otimes A \to \mathcal{B}_0(\mathcal{H}) \otimes A \otimes A: x \mapsto x \otimes 1$$ to a map $$M(\mathcal{B}_0(\mathcal{H}) \otimes A) \to M(\mathcal{B}_0(\mathcal{H}) \otimes A \otimes A)$$

Here, the tensor product is the minimal one.

Does every map $*$-morphism $A \to B$ between $C^*$-algebras extend to a $*$-morphism $M(A) \to M(B)$?

Thanks in advance for any reference/input/links.