# Confusion with $CW$ approximation

Hatcher give the following defintion of $CW$ approximation(page 352):

Given a pair $(X,A)$ where the subspace $A\subset X$ is a nonempty $CW$ complex, an n-connected CW model for $(X,A)$ is an $n$-connected $CW$ pair $(Z,A)$ and a map $f:Z\rightarrow X$ with $f|A$ the identity, such that $f_{*}:\pi_{i}(Z)\rightarrow \pi_{i}(X)$ is an isomorphism for $i>n$ and an injection for $i=n$, for all choices of basepoint. Since $(Z,A)$ is $n$-connected, the map $\pi_{i}(A)\rightarrow \pi_{i}（Z）$ is an isomorphism for $i<n$ and a surjection for $i=n$.

In the critical dimension $n$, the maps $A\rightarrow Z\rightarrow^{f}X$ induce a composition $\pi_{n}(A)\rightarrow \pi_{n}(Z)\rightarrow \pi_{n}(X)$ factoring the map $\pi_{n}(A)\rightarrow \pi_{n}(X)$ as a surjection followed by an injection.

My question is: Why $f_{*}$ must be an injection at $i=n$ for the above system to work? Can I require $i=n$ this be an isomorphism as well? I understand the need to introduce $Z$ is for the convenience of factoring the map, but I feel if I can make $\pi_{n}(Z)=\pi_{n}(A)$ the construction may actually be easier.

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