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In line 6, there is an explanation, '$h$ is an equivalence in $E$ if and only if $h$ is an equivalence in $C$. '

Suppose that $h : B \rightarrow D$ is an equivalence in $C$. Then $h$ is an equivalence in $E$, that is, $h : \left (B,\left \{ f_i \right \}\right)\rightarrow \left ( D, \left \{ g_i \right \} \right )$ is an equivalence in $E$. But if $Hom(B, A_i)=\phi$ for some $i\in I$, then $h : \left (B,\left \{ f_i \right \}\right)\rightarrow \left ( D, \left \{ g_i \right \} \right )$ doesn't make sense(In fact, $\left (B,\left \{ f_i \right \}\right)$ is not an object of $E$).

I wonder if you understand what I'm asking.. It seems like an unnecessary question but I cannot proceed. Can I get any advice?

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You will get more answers when you explain the notation. Not everyone has a copy of that book. –  Martin Brandenburg Feb 2 at 17:22
2  
If at some point you end up writing "I wonder if you understand what I'm asking", then you might consider rewriting the question... –  Pece Feb 2 at 17:51

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