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.