$*$ homomorphism $\phi$ from $A$ to multiplier algebra $M(I)$

Question

If $I$ is a closed ideal in $C^*$ algebra $A$, then there is a unique $*$ homomorphism $\phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $I\to M(I)$, where $M(I)$ is the multiplier algebra of $I$.

My question is:If $A$ has a unit $e$, the map $\phi$ must not be zero since $e \mapsto (L_{e},R_{e})$.

If $A$ is not unital, can we conclude that $\phi$ is also a nonzero $*$ homomorphism?

Answer 1

If $\phi$ is zero, then also $I \to M(I)$