Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ preserves and reflects weak equivalences and fibrations. It follows that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$, where $\mathrm{I}$ is a set of generating cofibrations of $\mathcal{C}$.

Now, assume that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$ and that $U$ reflects fibrant objects (or even all fibrations). Is the model structure on $\mathcal{D}$ is necessarily transferred in this case? If these conditions hold and the transferred model structure exists, then it necessarily coincides with the given one. Thus, the question can be reformulated as follows. Is there a model category $\mathcal{D}$ satisfying conditions given above such that the transferred model structure on $\mathcal{D}$ does not exist?

Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ preserves and reflects weak equivalences and fibrations. It follows that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$, where $\mathrm{I}$ is a set of generating cofibrations of $\mathcal{C}$.

Now, assume that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$ and that $U$ reflects fibrant objects (or even all fibrations). Is the model structure on $\mathcal{D}$ necessarily transferred in this case? If these conditions hold and the transferred model structure exists, then it necessarily coincides with the given one. Thus, the question can be reformulated as follows. Is there a model category $\mathcal{D}$ satisfying conditions given above such that the transferred model structure on $\mathcal{D}$ does not exist?