# ag.algebraic geometry – Berthelot-Ogus comparison isomorphism

On the following link, https://perso.univ-rennes1.fr/matthieu.romagny/exposes/Theoremes_de_comparaison_et_representatons_galoisiennes.pdf, page, $$2$$, the Berthlot-Ogus isomorphism theorem is stated as follows,

We have a canonical isomorphism, $$rho_{mathrm{cris}} : H_{mathrm{cris}}^{i} (X) otimes_{K_ {0}} K to H_{mathrm{dR}}^{i} (mathscr{X}) otimes_{O_K} K = H_{mathrm{dR}}^{i} (X) qquad (1)$$
On the other hand, on the following pdf, https://www.jmilne.org/math/Books/DMOS.pdf, page, $$359$$, the author claims that there is an isomorphism, $$H_ {mathrm{DH}} (X) otimes W simeq H_{mathrm{cris}} (X (s) / W) qquad (2)$$.

What is the difference between the two isomorphisms $$(1)$$ and $$(2)$$ ?

Are the two isomorphisms the same ?

Thanks in advance for you help.