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 $A$.

**question**

Is it true that $L(a)\sqcup_{L(b)} L(c)$ is a homotopy pushout in $B$ ?

derivedfunctor of $L$ on these object. $\endgroup$1more comment