A question from Fujino's book

Question

In Fujino's book Foundations of the minimal model program, the definition of semi-ample $\mathbb{R}$-divisors is as follows

Then he claims that

I don't understand the last sentence of the proof. Why is $f$ the morphism that has the property $D\sim_{\mathbb{R}}f^*A$? It seems to me that we only have $\sum_i D_i\sim_{\mathbb{R}}f^*A$

Also, is the proof saying that the "morphism" $f$ in Lemma 2.1.11(2) is in fact a contraction morphism?