It may be easier to prove that the map induced on the mapping spaces between fibrant and cofibrant objects is a weak equivalence, provided that $C$ and $D$ are enriched over some closed symmetric monoidal model category and $F$ is continuous. Of course this is not a necessary condition, but it may be considered a "homotopy coherent way" to say that $F$ is fully faithful on the level of the homotopy categories.

The definition of a model category is self-dual. So are all the general theorems, therefore there is no need to write a separate exposition of the theory of fibranly generated model categories. But the set-theoretical properties of the underlying category are not dualizable. More specific, if the original category os locally presentable, then its dual is almost never locally presentable (except for complete latices, see Adamek-Rosicky's book "Locally presentable and accessible categories"). In particular $\lambda$-presentable objects in a locally presentable model category are $\lambda$-copresentable in the dual category, but need not be presentable for any cardinal. Dualizing the standard theorems on Bousfield localization you obtain that in a dual of a combinatorial model categories there exist right Bousfield localizations (colocalizations) with respect to any set of maps and left Bousfield localizations with respect to any set of objects.

An enrichment over a closed sym. mon. model category is a self-dual concept, the roles of the tensor product and the cotensor product are switched in the opposite category.

It may be easier to prove that the map induced on the mapping spaces between fibrant and cofibrant objects is a weak equivalence, provided that $C$ and $D$ are enriched over some closed symmetric monoidal model category and $F$ is continuous. Of course this is not a necessary condition, but it may be considered a "homotopy coherent way" to say that $F$ is fully faithful on the level of the homotopy categories.

The definition of a model category is self-dual. So are all the general theorems, therefore there is no need to write a separate exposition of the theory of fibranly generated model categories. But the set-theoretical properties of the underlying category are not dualizable. More specific, if the original category is locally presentable, then its dual is almost never locally presentable (except for complete latices, see Adamek-Rosicky's book "Locally presentable and accessible categories"). In particular $\lambda$-presentable objects in a locally presentable model category are $\lambda$-copresentable in the dual category, but need not be presentable for any cardinal. Dualizing the standard theorems on Bousfield localization you obtain that in a dual of a combinatorial model categories there exist right Bousfield localizations (colocalizations) with respect to any set of maps and left Bousfield localizations with respect to any set of objects.

An enrichment over a closed sym. mon. model category is a self-dual concept, the roles of the tensor product and the cotensor product are switched in the opposite category.