We know that and adjoint pair $F\dashv G$ between two categories $\textsf{C}$ and $\textsf{D}$ is equivalent to give a natural transformation $\eta:1_{\textsf{C}}\to GF$ with universal property, and in the same fashion $\varepsilon:FG\to1_{\textsf{D}}$. But there is also a fourth equivalent condition which is giving me some trouble. Indeed $F\dashv G$ is also equivalent to ask for natural transformations $\eta$ and $\varepsilon$ to exist as above, but with the universal property replaced by triangular identities:
$$G\varepsilon_A\eta_{GA}=1_{GA}\qquad\varepsilon_{FB}F\eta_{B}=1_{FB}$$
(I'm not able to draw the triangles properly on latex).
I'm not really sure on how to define the bijection $$\varphi_{AB}:\textsf{D}(B, GA)\to\textsf{C}(FB,A)$$ The only idea I came up with is: $$\varphi_{AB}(v)=\varepsilon_A\circ Fv$$ and as candidate inverse: $$\varphi^{-1}_{AB}(u)=Gu\circ \eta_{B}$$ But I'm not really sure on how to use triangular identities in order to show this definition works, assuming is the right one. If we take $A=FB$ then it clearly works, as well as taking $B=GA$ but there's no reason for a generic object $A$ in $\textsf{C}$ to be written as $FB$ for a certain $B$ in $\textsf{D}$ and viceversa.