Kerodon

Proof. Let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is a $\kappa $-small transfinite composition of morphisms of $W$; we wish to show that $f \in W$. Choose an ordinal $\alpha < \kappa $ and a functor $X: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a $\kappa $-small transfinite composition of morphisms of $W$. For each $\beta \leq \alpha $, the functor $X$ carries the ordered pair $(0 \leq \beta )$ to a morphism $f_{\beta }: X(0) \rightarrow X(\beta )$. We will complete the proof by showing that each of the morphisms $f_{\beta }$ belongs to $W$. The proof proceeds by transfinite induction on $\beta $. If $\beta = 0$, then $f_{\beta } = \operatorname{id}_{X}$ and the desired result follows from assumption $(1)$. If $\beta $ is a nonzero limit ordinal, then the desired result follows from assumption $(3)$. It will therefore suffice to treat the case where $\beta = \gamma +1$ is a successor ordinal. In this case, the desired result follows by applying assumption $(2)$ to the diagram

since $f_{\gamma }$ belongs to $W$ by virtue of our inductive hypothesis. $\square$