In the category of C-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is this still true in the category of C-algebras and completely positive maps?

Suppose $f \colon A \to B$ is a completely positive monomorphism. If $f(a)=0$ for some positive $a \in A$, then $fg=fh$ for the completely positive maps $g,h \colon \mathbf{C} \to A$ defined by $g(z)=0$ and $h(z)=za$, whence $a=0$. What if $f(a)=0$ for an arbitrary $a \in A$? If we could still conclude $a=0$ then $\ker(f)=0$ and $f$ would be injective.