Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book *Model categories*, there is a model category structure on the diagram category $C^B$ such that weak equivalences and fibrations are defined pointwise.


Assume that $C\leftrightarrows D$ is a Quillen equivalence. By pointwise application, we obtain a pair of functors $C^B\leftrightarrows D^B$. Is this also a Quillen equivalence?



Quillen equivalence of diagram categories
Edit: As Fernando pointed out, the claim can be found in Hirschhorn's book as Proposition 15.4.1.


Here is my own attempt. We write $(F,G)\colon C\leftrightarrows D$ for the given Quillen equivalence and $(F^X,G^X)\colon C^X\leftrightarrows D^X$ for the pointwise induced adjunction. First we check that $(F^X,G^X)$ is a Quillen adjunction. It suffices to show that $G^X$ preserves fibrations and trivial fibrations which is clear because of the pointwise definition of fibrations and weak equivalences. It remains to show that $(F^X,G^X)$ is a Quillen equivalence, that is, a map $F^X(c^X)\to d^X$ in $D^X$ with $c^X$ cofibrant and $d^X$ fibrant is a weak equivalence if and only if the adjoint map $c^X\to G^X(d^X)$ is a weak equivalence in $C^X$. This follows from the pointwise definition of weak equivalences and fibrations once we know that cofibrant objects in $C^X$ are pointwise cofibrant (which can be shown inductively similarly to the proof of Proposition 6.8 in *DIAGRAM SPACES AND SYMMETRIC SPECTRA* by STEFFEN SAGAVE AND CHRISTIAN SCHLICHTKRULL).


