Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow B\oplus\pi(B)$ by $$\delta(b)=[\sigma\circ\pi(b)]\oplus\pi(b).$$ Then, can we verify that $\delta(B)\cong\pi(B)$?
Here is the definition of homotopic, which comes from the book "C*-algebras and Finite-Dimensional Approximations":
Definition 7.3.3. Let $\sigma_{0}:A\rightarrow B$ and $\sigma_{1}: A\rightarrow B~$ be *-homomorphisms. Then we say $\sigma_{0}$ and $\sigma_{1}$ are homotopic if there exist *-homomorphisms $\hat{\sigma}_{t}: A\rightarrow B$, $0\leq t \leq1$, such that $\hat{\sigma_{0}}=\sigma_{0}$, $\hat{\sigma_{1}}=\sigma_{1}$ and for every fixed $a\in A$ the map $[0, 1]\rightarrow B$, $t\rightarrow \hat{\sigma_{t}}(a)$, is continuous from the usual topology on $[0, 1]$ to the norm topology on $B$.