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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 $A$.

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

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    $\begingroup$ Have you seen: math.stackexchange.com/questions/138946/… $\endgroup$ Feb 8 '17 at 1:19
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    $\begingroup$ See also Rezk's comment here: mathoverflow.net/questions/82813/… . This seems to show that the answer to your question is "no", since Ch(R) is certainly left proper. $\endgroup$ Feb 8 '17 at 1:24
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    $\begingroup$ @DavidWhite I don't see how do you "conclude" that the answer is "no" following your links. $\endgroup$ Feb 8 '17 at 8:10
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    $\begingroup$ @DavidWhite My comment in that question was about a slightly different issue (which has to do with the use of the phrase "homotopy colimit diagram"; it is possible to have a "homotopy pushout square" in which none of the sides are cofibrations). $\endgroup$ Feb 8 '17 at 15:26
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    $\begingroup$ 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. $\endgroup$ Feb 8 '17 at 15:37

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