Clarification in Yoneda Lemma exercise from Aluffi's Algebra Chapter $0$

Question

I don't understand the sentence "Map $h_X$ to the image of $\mathrm{id}_X \in h_x(X)$ in $\mathscr F(X)$."

$h_X$ is a functor, so, what can that sentence mean?

Answer 1

You're right, this appears to be a typo. Rather than "Map $h_X$ to ...", it should say "Map this natural transformation to...".

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More precisely, the exercise is asking you to show that there is a bijection between the sets $\text{Nat}(h_X,\mathscr{F})$ and $\mathscr{F}(X)$. Then the author is giving you a hint by telling you how to define the bijection: Map a natural transformation $\alpha\in \text{Nat}(h_X,\mathscr{F})$ to $\alpha_X(\text{id}_X)\in \mathscr{F}(X)$, where $\alpha_X\colon h_X(X)\to \mathscr{F}(X)$ is the component of the natural transformation $\alpha$ at the object $X$.

Answer 2

It should read something like "map $\alpha: h_X\to \mathcal F$ to the image of $\mathrm{id}_X$..."