  A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all
$C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $\pi \colon
E \rightarrow B$ and for every $^*$-homomorphism $\phi \colon A \rightarrow E$,
any path of $^*$-homomorphisms $A \rightarrow B$ starting at $\pi \phi$ lifts
to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $\phi$. Blackadar
has shown that this property holds for all semiprojective $C^*$-algebras.
  We show that a version of the homotopy lifting property for asymptotic
morphisms holds for separable $C^*$-algebras that are sequential inductive
limits of semiprojective $C^*$-algebras. It also holds for any separable
$C^*$-algebra if the quotient map $\pi$ satisfies an approximate decomposition
property in the spirit of (but weaker than) the notion of quasidiagonality for
extensions.
