# Functor factorization theorem and the structure of a functor

Corollary 4.8 in Awodey's book states that every functor $$\mathcal F$$:$$\mathcal C\rightarrow\mathcal D$$ factors $$\mathcal F$$ = $$\mathcal H$$$$\mathcal G$$ where

$$\mathcal G$$ : $$\mathcal C\rightarrow\mathcal C/ker(F)$$ is bijective on objects quotient functor and $$\mathcal H$$ : $$\mathcal C/ker(F)\rightarrow\mathcal D$$ is fully faithful. That is the composite ;

$$\mathcal F$$:$$\mathcal C\rightarrow\mathcal C/ker(F)\rightarrow\mathcal D$$.

We can also factorize $$\mathcal H$$ in the way $$\mathcal F$$ was factorized to obtain $$\mathcal H$$=$$\mathcal H_1$$$$\mathcal G_1$$ initializing at $$\mathcal H_0$$= $$\mathcal H$$ and $$\mathcal G_0$$= $$\mathcal G$$.

Question: upon iterated factorization of $$\mathcal H_n$$, does this finally converge on $$\mathcal D$$. And if so, what factorization system generally captures this ? My guess would be reflective-factorizaton systems or monadic decompositon but am looking for a grounded answer.

(i) In the factorization $$\mathcal H$$=$$\mathcal H_1$$$$\mathcal G_1$$, $$\mathcal H_1$$=$$\mathcal H$$ and $$\mathcal G_1$$ is the identity on $$\mathcal C/ker(F)$$
(ii) An orthogonal factorization system ($$\mathcal L$$, $$\mathcal R$$) where the class $$\mathcal L$$ is of iterated strict-localizations(= iterated quotients) and $$\mathcal R$$ is the class of conservative functors. It is the factorization of a functor where the iterated quotient converges at the colimit of the transfinite sequence. See relevant links in comments.
• In the factorization $\mathcal{H}=\mathcal{H}_1\circ\mathcal{G}_1$, isn't $\mathcal{H}_1=\mathcal{H}$ and $\mathcal{G}_1$ the identity functor on $\mathcal{C}/\ker(F)$? Mar 25 at 11:57
• In the Corollary 4.8 that you cite, I see only that the functor you call $\mathcal H$ is faithful, not necessarily fully faithful. Apr 4 at 15:56