I recently found a very convincing and elegant proof of Q1 in Balzin's Reedy model structures in families. In the paper, it appears as Cor. 3.41.

The idea of the proof is that it is enough to show that the infinity category associated to a model category $\mathcal M$ has $\infty$-colimits over direct Reedy diagrams, because they imply the existence of pushouts and coproducts. (Dually for limits). In order to do so he uses that when $R$ is direct Reedy, $\mathcal M^R$ has always a model structure (without any additional assumption on $\mathcal M$) in order to build the Quillen adjunction, $$ \text{colim}: \mathcal M^R \leftrightarrows \mathcal M : c_R.$$ To finish he applies on of the main theorems of his machinary to derive the behaviour of the derived functors, Thm. 3.31 in the paper.

I recently found a very convincing and elegant proof of Q1 in Balzin's Reedy model structures in families. In the paper, it appears as Cor. 3.41.

The idea of the proof is that it is enough to show that the infinity category associated to a model category $\mathcal M$ has $\infty$-colimits over direct Reedy diagrams, because they imply the existence of pushouts and coproducts. (Dually for limits). In order to do so he uses that when $R$ is direct Reedy, $\mathcal M^R$ has always a model structure (without any additional assumption on $\mathcal M$) in order to build the Quillen adjunction, $$ \text{colim}: \mathcal M^R \leftrightarrows \mathcal M : c_R.$$ To finish he applies one of the main theorems of his machinery to derive the behaviour of the derived functors, Thm. 3.37 in the paper.

I recently found a very convincing and elegant proof of Q1 in Balzin's Reedy model structures in families. In the paper, it appears as Cor. 3.41.

The idea of the proof is that it is enough to show that the infinity category associated to a model category $\mathcal M$ has $\infty$-colimits over directed Reedy diagrams, because they imply the existence of pushouts and coproducts. (Dually for limits). In order to do so he uses that when $R$ is directed Reedy, $\mathcal M^R$ has always a model structure (without any additional assumption on $\mathcal M$) in order to build the Quillen adjunction, $$ \text{colim}: \mathcal M^R \leftrightarrows \mathcal M : c_R.$$ To finish he applies on of the main theorems of his machinary to derive the behaviour of the derived functors, Thm. 3.37 in the paper.

I recently found a very convincing and elegant proof of Q1 in Balzin's Reedy model structures in families. In the paper, it appears as Cor. 3.41.

The idea of the proof is that it is enough to show that the infinity category associated to a model category $\mathcal M$ has $\infty$-colimits over direct Reedy diagrams, because they imply the existence of pushouts and coproducts. (Dually for limits). In order to do so he uses that when $R$ is direct Reedy, $\mathcal M^R$ has always a model structure (without any additional assumption on $\mathcal M$) in order to build the Quillen adjunction, $$ \text{colim}: \mathcal M^R \leftrightarrows \mathcal M : c_R.$$ To finish he applies on of the main theorems of his machinary to derive the behaviour of the derived functors, Thm. 3.31 in the paper.