Suppose we have a Quillen adjunction between model categories $$L:A\leftrightarrow B: R$$ such that $A$ is left proper model category. Let $a\leftarrow b\rightarrow c$ be a diagram in $A$ such that $b\rightarrow c$ is a cofibration then (as far as I understand) $a\sqcup_b c$ is a homotopy pushout in $A$.
Is it true that $L(a)\sqcup_{L(b)} L(c)$ is a homotopy pushout in $B$ ?
• Also, be careful: the conclusion can be true (I think it's true if $B$ is also left proper), but the values of the functor $L$ on $a,b,c$ or the push out might not coincide with the values of the derived functor of $L$ on these object. Feb 8 '17 at 15:37