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 have two questions. The first one is, when we say that a subfunctor for $G$ is given by subsets of each $Gc$, we are identifying each $Fc$ with its image via $\alpha_c$, right? So we would have uniqueness up to isomorphic objects.
Secondly, for the concrete exercise regarding $C(-,c)$, could you provide any hints on how to characterize such sets?
So far I've only noted that given $\mu: d' \rightarrow d$, we need the precomposition,
$$ \mu^{*}: f \in C(d,c) \mapsto f\mu \in C(d',c) $$
to restrict to $Fd$ and corestrict to $Fd'$. Thus, $\mu^*(Fd) \subseteq Fd'$, and so
$$ \bigcup_{\forall \mu: \ d' \to d \\ \forall f\in Fd} f\mu \subset Fd' $$
but I haven't got from there to any fruitful conclusions.
In the last formula in the question, underneath the big union symbol, it should not say $f:d\to c$ but rather $f\in Fd$. With that correction, I think you've essentially finished the job. Another way to state the result is that the family $\bigcup_dF(d)$ should be closed under right-composition with arbitrary morphisms $\mu$ (whenever the composition makes sense). Such a family is called a sieve on $c$.