  We construct the crystalline comparison isomorphisms for proper smooth formal
schemes over an absolutely unramified base. Such isomorphisms hold for \'etale
cohomology with nontrivial coefficients, as well as in the relative setting,
i.e. for proper smooth morphisms of smooth formal schemes. The proof is
formulated in terms of the pro-\'etale topos introduced by Scholze, and uses
his primitive comparison theorem for the structure sheaf on the pro-\'etale
site. Moreover, we need to prove the Poincar\'e lemma for crystalline period
sheaves, for which we adapt the idea of Andreatta and Iovita. Another
ingredient for the proof is the geometric acyclicity of crystalline period
sheaves, whose computation is due to Andreatta and Brinon.
