Suppose that is a nice model category $\mathbf{M}$ (cofibrantly generated,...). Suppose we have a two diagrams $$F,G: \Delta^{op}\rightarrow \mathbf{M} $$ and a natural transformation $\nu: F\rightarrow G $ such that for any natural number $n$, $\nu_{n}: F(n)\rightarrow G(n) $ is a trivial cofibration. Is the induced map

$colim_{\Delta^{op}} F\rightarrow colim_{\Delta^{op}} G$ is a weak equivalence ?

injectivemodel structure on $M^{\Delta^{op}}$. But the colimit functor should only be expected to be left Quillen for theprojectivemodel structure (or maybe the Reedy model structure?). So presumably the answer isno, but I don't have a counterexample at hand. $\endgroup$ – Tim Campion Jun 28 '18 at 15:52