It has been proposed by Ettema and colleagues, in the two-domain framework for the tree of life, that Eukarya emerged from Heimdallarchaeia, as sister group to Hodarchaeales. Looking at the individual trees of the protein markers used by these authors, I notice that Eukarya are only sister to Hodarchaeales or other Heimdallarchaeia in a minority of trees, whereas they are located far apart from these Asgard archaea in most other trees. Examination of single trees also reveals massive gene transfers from Crenarchaeota and/or Korachaeota to hyperthermophilic Njordarchaeales, explaining why their belonging to Asgard archaea is sometimes difficult to recover. Finally, I discuss several points raised by Ettema and colleagues, such as the phylogeny of Asgard archaea and the hyperthermophilic nature of their last common ancestor. The patchy localization of Eukarya in individual trees relative to Hodarchaeales and other Heimdallarchaeia, as well as the patchy distribution of eukaryotic signature proteins among Asgard archaea, is best explained by suggesting that multiple gene transfers take place between proto-eukaryotes and Asgard archaea in both directions. This suggests that the co-evolution of proto-eukaryotes and Asgard archaea has played a major role in eukaryogenesis but also in shaping the physiology and diversification of Asgard archaea.