I am looking for an answer to the following question:
Let $F : {\cal C} \to {\cal D}$ be a left Quillen functor between combinatorial model categories; let $\tilde F \dashv \tilde G$ be the induced adjunction between the homotopy categories.
Which assumptions[1] on $F$ ensure that the comonad $\tilde F\tilde G : {\sf Ho}({\cal D})\to {\sf Ho}({\cal D})$ is a full functor?
[1] I already know that $F=1$ works. :-)
