Title: Injectivity of the connecting homomorphisms,
Abstract: Let $A$ be the inductive limit of a sequence $$A_1\, \xrightarrow{\phi_{1,2}}
\,A_2\,\xrightarrow{\phi_{2,3}} \,A_3\rightarrow\cdots$$ with
$A_n=\oplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$ are
Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms, in this paper,
we will prove that $A$ can be written as another inductive limit
$$B_1\,\xrightarrow{\psi_{1,2}} \,B_2\,\xrightarrow{\psi_{2,3}}
\,B_3\rightarrow\cdots$$ 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 $\phi_{n,n+1}$ are injective.