Weibel has shown that the filtered colimit functor $$ \varinjlim :(RMod)^I \rightarrow (RMod)$$ where $I$ is a filtered category, is exact. In corollary 2.6.16, pg 58 he claims
Corollarry 2.6.16. If $F:RMod \rightarrow B$, $B$ an abelian category. and $F$ is a left adjoint, then $L_*F(\varinjlim A_i) \simeq \varinjlim L_*F(A)$.
I am struggling to prove this:
Problem: How does one construct a projective resolution of $\varinjlim A_i$ that respects the preojctive resolution of $P_i \rightarrow A_i$?
I thought $\varinjlim P_i$ would be projective, but this turns out to be false as given in comments.