Injectivity of the connecting homomorphisms

Let $A$ be the inductive limit of a sequence $$A_1\stackrel{{\varphi }_{1,2}}{\to }{A}_2\stackrel{{\varphi }_{2,3}}{\to }{A}_3\to \cdots$$ with $A_n={\oplus }_{i=1}^{n_i}{A}_{[n,i]}$, where all the $A_{[n,i]}$ are Elliott-Thomsen algebras and ${\varphi }_{n,n+1}$ are homomorphisms, in this paper, we will prove that $A$ can be written as another inductive limit $$B_1\stackrel{{\psi }_{1,2}}{\to }{B}_2\stackrel{{\psi }_{2,3}}{\to }{B}_3\to \cdots$$ with $B_n={\oplus }_{i=1}^{n_i}{B}_{[n,i]}$, where all the $B_{[n,i]}$ are Elliott-Thomsen building blocks and with the extra condition that all the ${\varphi }_{n,n+1}$ are injective.