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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).

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).

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.

That us check the last claim. Assume $c^X$ is cofibrant, that is, the induced map $L_x c^X\to c^X_x$ is a cofibration for every $x\in X$, where $L_x$ denotes the latching space functor. An inductive argument shows that $c^X_x$ is cofibrant for every $x\in X$. (to be elaborated)

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).

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.

That us check the last claim. Assume $c^X$ is cofibrant, that is, the induced map $L_x c^X\to c^X_x$ is a cofibration for every $x\in X$, where $L_x$ denotes the latching space functor. To see that $c^X$ is pointwise cofibrant it is enough to check that $L_x c^X$ is cofibrant for every $x\in X$. I don't see that yet.

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.

That us check the last claim. Assume $c^X$ is cofibrant, that is, the induced map $L_x c^X\to c^X_x$ is a cofibration for every $x\in X$, where $L_x$ denotes the latching space functor. An inductive argument shows that $c^X_x$ is cofibrant for every $x\in X$. (to be elaborated)