Transfinite iteration of the generator extension preserves monomorphisms and factorizes small-source maps

For every successor ordinal, the transition map $M_{\alpha }\to M_{\alpha +1}$ is monic by [L1]. By transfinite induction, any transition map whose target is a successor stage is monic.

Let $\lambda$ be a limit ordinal and fix $\alpha <\lambda$. For $\alpha \le \beta <\lambda$, the short exact sequences $0\to M_{\alpha }\to M_{\beta }\to \mathrm{coker}(M_{\alpha }\to M_{\beta })\to 0$ form a filtered system. Exactness of filtered colimits under [L2, L4] makes the colimit sequence begin $0\to M_{\alpha }\to M_{\lambda },$ so the canonical map $M_{\alpha }\to M_{\lambda }$ is monic. Together with step 1.1, transfinite induction now shows that every transition map in the tower is monic.

By [L3], the subobjects of $U$ form a set; [A1] gives its cardinality $\kappa$. For each $N\subseteq U$, composition with $N\rightarrowtail U$ injects its subobject classes into those of $U$, so $\kappa$ bounds all their cardinalities. Fix $f:N\to M_{\lambda }$ with $N\subseteq U$, and regard each $M_{\alpha }$ as a subobject of $M_{\lambda }$ by step 2.1. The preimages $N_{\alpha }=f^{-1}(M_{\alpha })$ form an increasing family of subobjects of $N$, and [L2] gives ${\bigvee }_{\alpha <\lambda }{N}_{\alpha }=f^{-1}\text{\hspace{0.17em}⁣}\left({\bigvee }_{\alpha <\lambda }{M}_{\alpha }\right)=N\mathrm{.}$ For each distinct $N_{\alpha }$, take its least occurrence in the ordinal $\lambda$; these indices form a set $S\subseteq \lambda$ of size at most $\kappa$. Since $\mathrm{cf}(\lambda )>\kappa$, $S$ is bounded by some $\gamma <\lambda$. Then $N_{\gamma }$ contains every $N_{\alpha }$, so the displayed join gives $N_{\gamma }=N$. Equivalently, $f$ factors through $M_{\gamma }$.