Quick sanity check here:
Let $F: \mathcal{C} \to \mathcal{D}$ be a functor between two categories. Let $\mathcal{D}'$ be a subcategory of $\mathcal{D}$ such that $F(C) \in \mathcal{D}'$ for all $C \in Ob\mathcal{C}$ and $F(f) \in Hom_\mathcal{D'}(F(A),F(B))$ for every morphism $f \in Hom_\mathcal{C}(A,B)$.
Can we view $F$ as a functor $\mathcal{C}\to \mathcal{D}'$?
Attempt:
Yes, denote this functor with adapted codomain by $G$.
Our assumption implies that $G$ maps objects and morphisms to the correct place. Composition and identities are preserved by $G$ because they are preserved by $F$.
Is that correct?