Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors accessible, and all categories presentable (if needed, assume that $A \to C$ and $B \to D$ are left adjoints).

Suppose further that we have three saturated (that is, closed under retracts, pushouts and transfinite compositions) classes of arrows $S(B),S(C),S(D)$ of, respectively, $B,C,D$. Assume that the functors $B \to D, C \to D$ preserve these classes. (It seems likely to me that the set $S(D)$ should not play any importance in what follows)

Form the set $S(A)$ by requiring that it consists of all arrows of $A$ that are sent to $S(B)$ and $S(C)$ by the respective functors.

Question: if $S(B),S(C),S(D)$ are generated by (small) sets, does it imply that $S(A)$ is generated by a set as well?

EDIT: it may be not obvious that the formed class S(A) is weakly saturated. Assume it is.

Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors accessible, and all categories presentable (if needed, assume that $A \to C$ and $B \to D$ are left adjoints).

Suppose further that we have three saturated (that is, closed under retracts, pushouts and transfinite compositions) classes of arrows $S(B),S(C),S(D)$ of, respectively, $B,C,D$. Assume that the functors $B \to D, C \to D$ preserve these classes. (It seems likely to me that the set $S(D)$ should not play any importance in what follows)

Form the set $S(A)$ by requiring that it consists of all arrows of $A$ that are sent to $S(B)$ and $S(C)$ by the respective functors.

Question: if $S(B),S(C),S(D)$ are generated by (small) sets, does it imply that $S(A)$ is generated by a set as well?

EDIT: it may be not obvious that the formed class S(A) is weakly saturated. Assume it is.

EDIT2: I do not know if the original question admits an affirmative answer. What is true is that, following a remark of Tim to this post, one can consider the situation with the right classes. Let me leave a statement that may be of use for someone who googles this post up.

Lemma. Let $A$ be a presentable category together with a weak factorisation system $(L,R)$ having the property that

1. $(L,R)$ is functorial (a closer inspection may be used to drop this),
2. The functor $A^{[1]} \to A^{[2]}$ associated to the weak factorisation system is accessible,
3. The right class $R$ viewed as a subcategory of $A^{[1]}$ is accessible and accessibly embedded.

Then there exists a set $L_0 \subset L$ generating $(L,R)$.

Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors accessible, and all categories presentable (if needed, assume that $A \to C$ and $B \to D$ are left adjoints).

Suppose further that we have three saturated (that is, closed under retracts, pushouts and transfinite compositions) classes of arrows $S(B),S(C),S(D)$ of, respectively, $B,C,D$. Assume that the functors $B \to D, C \to D$ preserve these classes. (It seems likely to me that the set $S(D)$ should not play any importance in what follows)

Form the set $S(A)$ by requiring that it consists of all arrows of $A$ that are sent to $S(B)$ and $S(C)$ by the respective functors.

Question: if $S(B),S(C),S(D)$ are generated by (small) sets, does it imply that $S(A)$ is generated by a set as well?

Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors accessible, and all categories presentable (if needed, assume that $A \to C$ and $B \to D$ are left adjoints).

Suppose further that we have three saturated (that is, closed under retracts, pushouts and transfinite compositions) classes of arrows $S(B),S(C),S(D)$ of, respectively, $B,C,D$. Assume that the functors $B \to D, C \to D$ preserve these classes. (It seems likely to me that the set $S(D)$ should not play any importance in what follows)

Form the set $S(A)$ by requiring that it consists of all arrows of $A$ that are sent to $S(B)$ and $S(C)$ by the respective functors.

Question: if $S(B),S(C),S(D)$ are generated by (small) sets, does it imply that $S(A)$ is generated by a set as well?

EDIT: it may be not obvious that the formed class S(A) is weakly saturated. Assume it is.