Right adjoint unique up to isomorphism i want to prove the following without the Yoneda Lemma (because it is the exercise): Suppose $F\dashv G$ (with unit $\eta$ and counit $\epsilon$) and $F\dashv G'$ (with unit $\eta'$ and conunit $\epsilon'$) then $G\cong G'$.
I want to do this in three steps:

*

*Construct $G(X)\rightarrow G'(X)$ and $G'(X)\rightarrow G(X)$ with help of the units and counits.

*Show that this arrows are natural transformations.

*Both arrows are each other inverse.


This is may plan so far. I will tell you what i have done so far and what my problems are:

*

*I used unit and counit to make $f:=G(X)\rightarrow G'(X)$ by $f_X=G'(\epsilon_X)\circ\eta_{G(X)}$. This is form my point of view a corollary of the triangle identity. On the same way we can define $g_X:=G'(X)\rightarrow G(X)$ by $g=G(\epsilon'_X)\circ\eta_{G'(X)}$.

My Question: Is this construction okay? If so then we can go to point 2, if not tell me why not and maybe you can help me further by correcting the following mistakes?

*

*Okay, second step. I have to show that if $h:X\rightarrow Y$, that then the following holds: $G'(h)\circ f_X=f_Y\circ G(h)$.

My problem: i do not see how to came form the one side of the equation to the other side. Is this trivally or what for computations must be done?

*

*I can't to the second step without help, so i can not do the third step. Can someone help me with this step, too?


I am very happy about each sort of help, information and solution.
Thank you for reading and thinking about that. Thank you also for help.
 A: *

*Yes, it seems ok.

*Prove more generally that the constructions $F\alpha:=x\mapsto F(\alpha_x)$ and $\alpha G:=x\mapsto \alpha_{Gx}$ give natural transformations if $F,G$ functors and $\alpha$ is a natural transformation, and that composition of natural transformations ($\alpha\circ\beta:=x\mapsto \alpha_x\circ\beta_x$) is again natural. 
So that, now we have the following composition of natural transformations:
$$\varphi: G \ \overset{\eta'G}\longrightarrow\  G'FG \ \overset{G'\varepsilon}\longrightarrow\ G' $$
(what you called $f$).

*We have $\varphi=G'\varepsilon\circ \eta'G$ and $\psi=G\varepsilon'\circ\eta G'$. 


My favorit method is to draw squares for the units and counits with edges $1_{\Bbb A}$, $1_{\Bbb B}$ (omitted) and $F$, $G$ as below, and paste them together in appropriate ways.
$$ \matrix{&\overset{G'}\longrightarrow  \\
&\ \ \ \ \ \ \  \varepsilon'\  \downarrow F \\
\ } \quad \quad
\matrix{&&  \\
F\downarrow\ \eta  \\ \phantom{F\downarrow} \underset{G}\longrightarrow
} 
$$
The squares are read from top right to bottom left, corresponding to e.g. $\varepsilon':FG'\to 1_{\Bbb B}$.
Then consider the following pasting of squares:
$$\matrix{\varphi & : &\varepsilon& \eta' & 1_{1_{\Bbb A}} \\
\psi & : &1_{1_{\Bbb B}} & \varepsilon' & \eta}$$
