  We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$
to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a
manner reminiscent of several tracial approximation properties. We are
particularly interested how regularity properties in the Elliott classification
program pass from $B$ to $A$. Among them, we show that the strict comparison
property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.
