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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).