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For a smooth projective variety $X$ defined over $k$ which admits a real embedding $\sigma:k \rightarrow \mathbb{C}$, its Betti cohomology is defined by \begin{equation} H^*_{B,\sigma}(X):=H^*(X \times_{k,\sigma}\mathbb{C}(\mathbb{C}),\mathbb{Q}) \,, \end{equation} where $X \times_{k,\sigma}\mathbb{C}(\mathbb{C})$ is the complex valued points of $X \times_{k,\sigma}\mathbb{C}$.

Since the embedding is real, complex conjugation acts on the points of $X \times_{k,\sigma}\mathbb{C}(\mathbb{C})$, which induces an involution $F_{\infty}$ on $H^*_{B,\sigma}(X)$.

The etale cohomology is defined by \begin{equation} H^*_{et}(X)_{\ell}:=H^*(X \times_k \bar{k},\mathbb{Q}_{\ell}) \,. \end{equation} There is a standard comparison isomorphism, $I_{\ell,\bar{\sigma}}$ \begin{equation} I_{\ell,\bar{\sigma}}:H^*_{B,\sigma}(X) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \simeq H^*_{et}(X)_{\ell} \end{equation} which depends on the choice of an extension of $\sigma$ to $\bar{\sigma}:\bar{k} \rightarrow \mathbb{C}$.

From lots of references, under this isomorphism, the involution $F_{\infty} \otimes 1$ corresponds to the automorphism $\bar{\sigma}^*(c) \in \text{Gal}(\bar{k}/k)$ (which acts on etale cohomology) where $c$ is complex conjugation which acts on $\bar{k}$ through the embedding.

Could anyone explain the ideas in the proof of this comparison isomorphism and the correspondence of the two involutions? Or give some references?

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I think you should first understand the comparison theorem in the case of smooth varieties over $\mathbb{C}$ for finite coefficients $\mathbb{Z}/n\mathbb{Z}$. For this you should consult SGA 4, Exposé XI.

Now given a smooth variety $X$ over $\bar{k}$ (using your notations) and an embedding $\sigma : \bar{k} \to \mathbb{C}$, there is a natural isomorphism between étale cohomology of $X$ and étale cohomology of $X_{\sigma}$ (the base change of $X$ to $\mathbb{C}$ using $\sigma$).

The result you want (the fact that complex conjugations match) follows from the functoriality of the two isomorphisms explained above. Let $c$ be the complex conjugation on $\bar{k}$ induced by the usual complex conjugation on $\mathbb{C}$. Given a smooth variety $X$ over $k$, there is a commutative diagram $$\require{AMScd} \begin{CD} X_{\bar{k}} @>>> X_\mathbb{C} \\ @VVV @VVV \\ X_{\bar{k}} @>>> X_\mathbb{C} \end{CD}$$ where the vertical arrows are given by $c$.

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