Showing the Naturality of Two (Isomorphic) Left Adjoints

Question

From pg. 85 of Categories for the Working Mathematician:

Problem: I understand everything except the red underlined portion.

To introduce some notation, let $h: x \rightarrow x'$ in $X$. Then showing that $\theta$ is natural is tantamount to showing that

$$ F'h \circ \theta_x = \theta_{x'} \circ Fh $$

Answer 1

Your last equation is an equality between two arrows $Fx\to F'x'$. Thus it is enough to check that their images under the bijection $A(Fx,F'x')\to X(x,GF'x')$ are equal. You know (for example from your previous question) that this bijection is given by $f\mapsto G(f)\circ \eta_x$, so it is enough to check that : \begin{align}G(F'h\circ \theta_x)\circ \eta_x & = GF'h\circ G\theta_x\circ \eta_x \\ & = GF'h\circ \eta_x'\\ & = \eta_{x'}'\circ h \\ & = G\theta_{x'}\circ \eta_{x'}\circ h \\ & = G\theta_{x'}\circ GFh\circ \eta_x \\ & = G(\theta_{x'}\circ Fh)\circ \eta_x.\end{align}