For category $\mathcal{C}$, does subclass $M \subseteq \operatorname{Mor}(\mathcal{C})$ closed under even largely wide pullbacks necessarily consist of only monomorphisms? The assumption being more specifically that for every even large sink with components in $M$, its pullback must exist in $\mathcal{C}$ and all its components must necessarily belong to $M$.
Indeed, often when considering e.g. orthogonal factorization systems one considers such subclass $M$ but one takes a priori assumption that the subclass consists of only monomorphisms: I wonder if this assumption is actually necessary?
The diagonal argument (or Lawvere's fixed point theorem) shows that if an object $Y$ has a weak power indexed by a class including the morphisms $X\to Y$, then there is at most one morphism $X\to Y$. In particular, $Y$ has weak powers indexed by all collections of morphisms $X\to Y$ if and only if it is a subterminal object, i.e. admits at most one morphism from any other object, in which case it is its own power.
Since wide pullbacks are products in the slice category, and monomorphisms are subterminal objects, it follows that a morphism having wide pullbacks with itself implies it is a monomorphism.