Exercise 4.8.8.2 (Uniqueness). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. The proof of Proposition 4.8.8.1 constructs a factorization
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}} \]
where $G$ is faithful and $F'$ is both full and bijective on objects. Show that these properties characterize the category $\operatorname{\mathcal{D}}'$ up to (unique) isomorphism.