Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the small object argument such that $LLP(RLP(M))$ is exactly the class of all monomorphisms of $X$. Recall that a separated segment (a separated interval) is a triple $(I,\partial^0,\partial^1)$ where $\partial^i:*\to I$ (where $*$ denotes the terminal object) and such that the pullback of the diagram $\partial^0:*\hookrightarrow I\hookleftarrow *:\partial^1$ is the empty presheaf.

This triple defines a functorial cylinder $(I\times(-),\partial^0\times id_{(-)}, \partial^1\times id_{(-)},\sigma\times id_{(-)})$ where $\sigma:I\to *$ is the terminal map. By abuse of notation, we will write for any object $P$ in $X$, $\partial^0_P:=\partial^0\times id_P, \partial^1_P:=\partial^1\times id_P$ and $\sigma_P:= \sigma\times id_P$. Since $X$ is a presheaf topos, we can see easily that given any monomorphism $K\to L$ in $X$, the square:

$$\begin{matrix}K&\hookrightarrow &L\\ \downarrow&&\downarrow\\ I\times K&\hookrightarrow&I\times L\end{matrix}$$

(where the vertical arrows are the induced maps $\partial^0_K$ and $\partial^0_L$, or $\partial^1_K$ and $\partial^1_L$) is cartesian and is composed exclusively of monomorphisms. Because of this very fine property, we may define $I\times K \cup \{i\}\times L$ to be the subobject of $I\times L$ given by the apparent inclusion of the pushout where $i$ depends on the $\partial^i$ appearing in the above diagram. Adding to our list of suggestive notation, we define the map $\partial I:= *\coprod *$ considered as a subobject of $I$ by the canonical map from the coproduct $(\partial^0,\partial^1)$ (similarly, we define $\{0\}$ and $\{1\}$ to be the subobjects corresponding to the obvious maps (in this notation, $\partial I = \{0\}\coprod \{1\}$). We denote the previously mentioned inclusion $(\partial^0,\partial^1)$ by $b:\partial I\hookrightarrow I$, and as with the other distinguished maps, putting a subscript gives the obvious piece of the natural transformation.

Given any two morphisms $f:A\to A',g:B\to B'$ in $X$, define their smash product $f\wedge g:A\times B'\coprod_{A\times B} A'\times B \to A'\times B'$. Note that the smash product gives a monoidal product on $Arr(X)$ (the unit being the inclusion of the empty presheaf into the terminal one).

Given a separated segment $I$ in $X$, define a class of anodyne morphisms relative to $I$ to be a class $An$ of monomorphisms of $X$ satisfying the following three conditions:

$An_0:$ There exists a small set $S$ of monomorphisms such that $An=LLP(RLP(S))$.

$An_1:$ For any monomorphism $f:K\hookrightarrow L$, the smash products $\partial^i\wedge f$ are elements of $An$ for $i=0,1$.

$An_2:$ For any $f\in An$, the smash product $b\wedge f$ is an element of $An$ (recall again that $b:\partial I\to I$ is the canonical inclusion).

Question: Given any category of presheaves $X$, any separated segment $I$ on $X$, and any class of morphisms $An$ anodyne with respect to $I$, is it the case that given any monomorphism $f$ in $X$ and any anodyne morphism $g$ in $An$ that $f\wedge g\in An$? If this is true would you mind sketching a proof?

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