I am studying Proposition 3.7.4 of Francis Borceux's Handbook of Categorical Algebra. For specific functors $F,G:\mathcal{B}\rightarrow\mathcal{D}$, he shows that for any functor $H:\mathcal{B}\rightarrow\mathcal{D}$, we always have the bijection $$\textbf{Nat}(F,H)\cong\textbf{Nat}(G,H)$$ He then claims that by taking $H = F$ for which we find $\textbf{Nat}(F,F)\cong\textbf{Nat}(G,F)$ and then $H = G$ for which we find $\textbf{Nat}(F,G)\cong\textbf{Nat}(G,G)$, it follows that $F\cong G$.
I do not see why this is true? Is this a general fact, or does Borceux silently depend on the chosen bijection between the sets of natural transformations?