I have recently been stumped on the following exercise from Weibel's text "An Introduction to Homological Algebra." It is Exercise 8.4.5: Suppose that $\mathcal{A}$ has enough projectives, so that the category $\mathcal{SA}$ of simplicial objects in $\mathcal{A}$ has enough projectives. Show that a simplicial object $P$ is projective in $\mathcal{SA}$ if and only if (1) each $P_n$ is projective in $\mathcal{A}$, and (2) the identity map on $P$ is simplicially homotopic to the zero map.
For the sake of simplicity and familiarity, I have set $\mathcal{A}$ to be the category of modules over a ring. I am really only concerned with the "if" direction. I had done a similar exercise, but for the category of chain complexes, and not simplicial objects (Weibel Exercise 2.2.1). For $$f:B\longrightarrow C$$ onto, and any map $$g:P\longrightarrow C,$$ there exists morphisms $t_n:P_n\longrightarrow B_n$ such that $f_nt_n=g_n$ for all $n$.
I had built my chain map $\gamma:P\longrightarrow B$ by setting $\gamma_n:=d_{n+1}^Bt_{n+1}h_n+t_nh_{n-1}d_n^P$ (where the map $h$ is my retracting map). Then one can check that this satisfies everything. I figured this would work for the simplicial modules as well, but I don't think it does. For the simplicial modules, we would need $$\gamma_{n-1}\delta_i^P=\delta_i^B\gamma_n$$ for all $n\geq0$ and $0\leq i\leq n$, which is asking quite a bit more.
Does anyone have any hints or ideas about how to solve this exercise? Or any ideas about how to construct the appropriate map?