Projective objects in the category of simplicial objects in an abelian category

Let $$S\mathcal{A}$$ be the category of simplicial objects in an abelian category $$\mathcal{A}$$. In exercise 8.4.5 in Weibel's An Introduction to Homological algebra, it is said that $$P \in S\mathcal{A}$$ is projective if each $$P_i$$ is projective and $$P$$ is null-homotopic.

By Dold-Kan correspondence we can pass to the category of chain complex $$C\mathcal{A}_{\ge0}$$. Consider $$P_{\cdot}$$ s.t. $$P_0$$ is projective and $$P_i=0$$ for $$i\neq 0$$. Consider $$B_\cdot \rightarrow A_\cdot$$ which is surjective and $$P_\cdot \rightarrow A_\cdot$$. Obviously $$B_0 \rightarrow A_0$$ is surjective and we can lift $$P_0 \rightarrow A_0$$ to $$P_0\rightarrow B_0$$. This gives $$P_\cdot \rightarrow B_\cdot$$ which is a lifting. So such $$P_\cdot$$ is projective but not null-homotopic.

Am I wrong or Weibel? If Weibel is wrong, what is the proper characterisation of projective simplicial objects?