Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-algebras for the Choi-Effros product.

Given a unital $*$-morphism $f: A \to B$, is it true that there exists a unique unital $*$-morphism $\overline{f}: I(A) \to I(B)$ that extends $f$?

Once the above question is answered positively (if the answer is positive), the following will probably be easy:

Is this construction functorial? I.e. is $I(-)$ a functor from the category of unital $C^*$-algebras to the category of unital $C^*$-algebras (with morphisms unital $*$-homomorphisms?

A reference is more than enough for me to be satisfied with an answer.