There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained [here](https://mathoverflow.net/a/92548/24563) and which is due to Sjoerd Crans.


The difficult part is to check the third condition. By [Axiomatic homotopy theory for operads](https://arxiv.org/abs/math/0206094) (2.6), it suffices to check that $\mathcal{N}$ has a functorial fibrant replacement and a functorial path-object for fibrant objects. This paper cites as references: 


1. D. G. Quillen, Homotopical algebra, Lect. Notes Math. 43 (1967) Theorem II.4
2. [Every homotopy theory of simplicial algebras admits a proper model](https://doi.org/10.1016/S0166-8641(01)00057-8), Theorem 7.6
3. [Algebras and modules in monoidal model categories](http://www.math.uni-bonn.de/people/schwede/AlgebrasModules.pdf), Theorem A.3


The first reference is about the construction of a model category structure on the category of simplicial objects of a category satisfying some conditions. The second reference is a "useful lemma" Lemma 7.6. As for the third reference, I just can't find where is Theorem A.3.



> 
> Could someone give a reference for the proof of this fact (or the proof) ?
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> 
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Transporting a model category structure along a left adjoint
It suffices to dualize the proof of Theorem 2.2.1 in [Necessary and sufficient conditions for induced model structures](https://arxiv.org/abs/1509.08154). It uses indeed an argument coming from Quillen's book "Homotopical Algebra", II page 4.9 (the diagram in the bottom part of the page). And it is also necessary to use the fact that the class of weak equivalences satisfies the 2-to-6 property.


