Given a pair of functors: $$F:\mathcal{B}\to\mathcal{A}\quad G:\mathcal{A}\to\mathcal{B}$$ Consider an identification: $$\alpha:F(B)\to A\leftrightarrow\beta:B\to G(A)$$ Then they're adjoints if the identification is natural.
What does natural mean in this context?
In most textbooks it means a natural transformation between two functors from $\mathcal{B}^{op}\times\mathcal{A}$ to sets, namely $Hom(F(\_),\_)$ and $Hom(\_,G(\_))$. The notation is not quite right but I hope it is clearer this way than if I wrote in the projection functors.
Really, though, you should also see Lawvere's version at https://en.wikipedia.org/wiki/Comma_category where it is not a natural transformation but an isomorphism of comma categories.
The proof that the two agree (modulo details of foundations) is worth working through just to understand what adjunction really is.