In Vakil's book "The Rising Sea", page 43-44.
Two covariant functors $F:\mathscr A\to \mathscr B$ and $G: \mathscr B\to \mathscr A$ are adjoint if there is a natural bijection for all $A\in \mathscr A$ and $B\in \mathscr B$ $$\tau_{AB} : \operatorname{Mor}_{\mathscr B}(F(A),B)\to \operatorname{Mor}_{\mathscr A}(A,G(B)).$$ We say that $(F,G)$ form an adjoint pair, and that $F$ is left-adjoint to $G$ (and $G$ is right-adjoint to $F$). We say $F$ is a left adjoint (and $G$ is a right adjoint). By "natural" we mean the following. For all $f: A\to A'$ in $\mathscr A$, we require $$ \require{AMScd}\begin{CD} \operatorname{Mor} _{\mathscr B}(F(A'),B)@>Ff^*>> \operatorname{Mor} _{\mathscr B}(F(A),B) \\ @VV \tau_{A'B}V @VV\tau_{AB}V \\ \operatorname{Mor} _{\mathscr A}(A',G(B))@>f^* >> \operatorname{Mor} _{\mathscr A}(A,G(B)) \end{CD} $$ to commute, and for all $g: B\to B'$ in $\mathscr B$ we want a similar commutative diagram to commute.
Is the meaning of the last sentence the following commutative diagram?
$$ \require{AMScd}\begin{CD} \operatorname{Mor} _{\mathscr A}(A,G(B))@>Gg^*>> \operatorname{Mor} _{\mathscr A}(A,G(B')) \\ @VV \tau_{AB}^{-1}V @VV\tau_{AB'}^{-1}V \\ \operatorname{Mor} _{\mathscr B}(F(A),B)@>g^* >> \operatorname{Mor} _{\mathscr B}(F(A),B') \end{CD} $$
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