For brevity let's call \$f_n\circ...\circ f_2\$ \$f\$
\$f\$ is surjective: for all \$e_{1,i}\$, \$f(e_{n,i})=e_{1,i}\$. \$e_{n,i}\$ exists.
\$f\$ is injective: suppose \$f(e_{n,i})=f(e_{n,j})\$. Then \$e_{1,i}=e_{1,j}\$, which can only happen if \$i=j\$.