I am reading Roman's new book "An Introduction to the Language of Category Theory" (2015). On p.60 he starts to describe the Yoneda Lemma. In short he says:
Let $C$ be a category and $a \in C$ is an object. Given a functor $H:C \longrightarrow Set$ and a natural transformation $\lambda: hom_C(a,-) \Longrightarrow H$. Then there is an element $p \in Ha$ such that $\lambda _x (g) = Hg(p)$ for all arrows $g:a \longrightarrow x$ in $C$. The element $p \in Ha$ "completely characterizes" the natural transformation $\lambda$.
I do not understand the last sentence, and I want to ask: 1) what does "completely characterizes" means in mathematics in general, and 2) what do I have to do to prove this statement of Roman?
I hope someone can help me with this.
Here is something that will hopefully help. If you plug in $a$ to the functors $hom_C(a,-)$ and $H$, then the natural transformation $\lambda$ gives a map
$$\lambda_a:hom_C(a,a)\rightarrow Ha$$
There is a natural element to consider in $hom_C(a,a)$, namely the identity map. Let $p = \lambda_a(id\rvert_a)$. Then write down the commutative square coming from $\lambda$, for an arrow $g:a\rightarrow x$ and you will see the property $\lambda_x(g) = Hg(p)$.
The sentence that $p$ completely characterizes the natural transformation $\lambda$, comes from the equation $\lambda_x(g) = Hg(p)$. That is, we can recover the natural transformation $\lambda$ from knowing $p$ in the following sense. Given $p$, we can define a natural transformation, $\hat{\lambda}:hom_C(a,-)\rightarrow H$ by the rule for $x$ in $C$ and $g: a\rightarrow c$, define
$$\hat{\lambda}_x(g) := Ha(p)$$
Then $\hat{\lambda}$ is the natural transformation $\lambda$. In this sense, $\lambda$ is determined by $p$.