I'm working through the following,
Exercise 2.1.iv. A functor $F$ defines a subfunctor of $G$ if there is a natural transformation $\alpha: F \Rightarrow G$ whose components are monomorphisms. In the case of $G: C^{op} → \operatorname{Set}$, a subfunctor is given by a collection of subsets $Fc \subset Gc$ so that each $Gf : Gc \longrightarrow Gc'$ restricts to a function $Ff: Fc → Fc'$. Characterize those subsets that assemble into a subfunctor of the representable functor $C(−, c)$.
I am wanting to know how do I fully characterize these subsets they speak of. I saw that this was already asked but it feels incomplete.
The person says that it is a collection of morphisms with codomain $c$ that are closed under precompositions with morphisms in $\textbf{C}$. My question is how do we prove that instead of show it like they did.