  We prove, under mild assumptions, that a Quillen equivalence between
symmetric monoidal model categories gives rise to a Quillen equivalence between
their model categories of (non-symmetric) operads, and also between model
categories of algebras over operads. We also show left properness results on
model categories of operads and algebras over operads. As an application, we
prove homotopy invariance for (unital) associative operads.
