Reference for Reedy weak factorization systems, not Reedy model structure?

Question

What paper/book is appropriate as the standard reference for Reedy weak factorization systems, not Reedy model structure? Specifically, I would like a reference for the following Propositions 1 and 2:

Proposition 1. Let $D = (D, D_-, D_+)$ be a small Reedy category. Let $C$ be a category with a weak factorization system $(L, R)$. Then we have the weak factorization system $(L^D, R^D)$ on $C^D$ given by the following, as long as $C$ has enough limits and colimits to make the following meaningful: $$ L^D := \left\{ X \to Y \text{ in } C^D \mathrel{}\middle|\mathrel{} \forall d \in D,\, L_dY \cup_{L_dX} X_d \to Y_d \text{ is in } L \right\} \\ R^D := \left\{ X \to Y \text{ in } C^D \mathrel{}\middle|\mathrel{} \forall d \in D,\, X_d \to Y_d \times_{M_dY} M_dX \text{ is in } R \right\} $$ Here, $L_dX, M_dY \in \operatorname{Ob} C$ denote the latching and matching objects of $X, Y : D \to C$, respectively, at $d \in D$.

Proposition 2. Let us work in the same setting as Proposition 1. Assume that every matching category is empty or connected, and that $C$ has all colimits of shape $D$. Then $\mathrm{colim} \colon C^D \to C$ maps morphisms in $L^D$ to $L$.

I need the case where $D = \{0 \leftarrow 1 \to 2\}$, for which I have the proof myself. However, despite my ignorance, I am almost sure that these propositions are elementary and well-known; for example, Hovey's Model Categories essentially states Proposition 1 in terms of model categories, but not in terms of weak factorization systems. However, I do not know what to cite for these propositions, because I do not know the result stating that the weak factorization system I am considering is a part of a model structure (specifically, inner anodyne morphisms in simplicial sets). I would greatly appreciate your suggestions.

Answer 1

The concept of Reedy weak factorization system itself appears in the following papers:

• Riehl E., Inductive Presentations of Generalized Reedy Categories, unpublished note, retrieved on Sep 9, 2024 from https://emilyriehl.github.io/files/generalized-reedy.pdf
• Shulman M., Reedy categories and their generalizations, arXiv:1507.01065v1.

In particular, see Definition 5.1 from Riehl, and Theorem 7.2 from Shulman.

However, as @john pointed out in comments, since a weak factorization system is equivalent to a model structure whose weak equivalences are all morphisms (modulo completeness), the construction of Reedy weak factorization systems follows directly from that of the Reedy model structures. So one could refer to and cite Hovey's classical Model categories, for example.