Firstly consider the four definitions in the question: How to define rigorously […].
Also consider the following definition:
Definition: Let $C,D$ be two categories and $F,G:[C]\to [D]$ be two functors. Suppose that $\alpha:F\to G$ is a morphism of functors
$F$ and $G$. We say that $\alpha$ is a functorial in $S$ if, for
all $T\in \text{Obj}(C)$ and $f\in \text{Hom}_C(T,S)$, the following
diagram commutes:
The book "Manifolds, Sheaves, and Cohomology" (written by Torsten Wedhorn) gives the following definition of adjoint functors:
Definition: Let $C,D$ be two categories and let $F:[C]\to [D]$ and $G:[D]\to [C]$ be functors. Then $G$ is said to be right adjoint
to $F$ and $F$ is said to be left adjoint to $G$ if for all
$X\in\text{Obj}(C)$ and $Y\in\text{Obj}(D)$ there is a bijection$$\text{Hom}_C(X,G(Y))\cong \text{Hom}_D(F(X),Y),$$
which is functional in $X$ and in $Y$.
Sincerely, I didn't understand the definition above. I tried to use a bijection $\Gamma:\text{Hom}_C(X,G(Y))\to \text{Hom}_D(F(X),Y)$ to construct a morfism of functors which is functorial in $X$ but I was unable to do this.
In view of the definitions of morphism of functors and functorial in a set, the definition above makes no sense to me.
MY QUESTION: What, possibly, did the author of that book mean by that definition?
Best Answer
A correction to start: you've copied the first definition incorrectly. $\alpha$ is not assumed to be a morphism of functors. Instead, $\alpha$ is assumed to be a family of morphisms (in $D$) $\alpha(S)\colon F(S)\to G(S)$, for all objects $S$ in $C$. If the family $\alpha$ is functorial in $S$, then we call $\alpha$ a morphism of functors $F\to G$.
Another comment here: What Wedhorn calls "functorial in $S$" is what most people would call "natural in $S$". A morphism of functors is often called a "natural transformation".
Now based on the very very brief introduction to categories and functors given in the pages leading up to the definition of adjoint functors, you're right to be confused at this point by what Wedhorn means when he writes that a bijection is "functorial in $X$ and $Y$". Here's what's going on:
Given a pair of functors $F$ and $G$ and objects $X$ in $C$ and $Y$ in $D$, we can consider the set $\text{Hom}_C(X,G(Y))$. If we fix $X$ and let $Y$ vary, we can check that we get a functor $\text{Hom}_C(X,G(-))\colon D\to \mathsf{Set}$.
Edit: More precisely, this functor sends an object $Y$ in $D$ to the set $\text{Hom}_C(X,G(Y))$. Given a morphism $\psi\colon Y\to Z$ in $D$, the functor $G$ gives us a morphism $G(\psi)\colon G(Y)\to G(Z)$ in $C$, and we can compose an arbitrary morphism $f\colon X\to G(Y)$ with $G(\psi)$ to get a morphism $G(\psi)\circ f\colon X\to G(Z)$. This is how the functor acts on morphisms: it sends $\psi\colon Y\to Z$ to the map of sets $\text{Hom}_C(X,G(Y))\to \text{Hom}_C(X,G(Z))$ given by $f\mapsto G(\psi)\circ f$.
On the other hand, if we fix $Y$ and let $X$ vary, then we get a functor $\text{Hom}_C(-,G(Y))\colon C^{\text{op}}\to \mathsf{Set}$. (Note the $\text{op}$! This is a contravariant functor from $C$ to $\mathsf{Set}$, with the action on morphisms $\psi$ given by precomposition with $F(\psi)$ instead of postcomposition.)
You can also think of $\text{Hom}_C(-,G(-))$ as a functor $C^{\text{op}}\times D\to \mathsf{Set}$, where the domain is the product category - but it's not necessary.
Similarly, $\text{Hom}_D(F(X),-)$ is a functor $D\to \mathsf{Set}$ for fixed $X$, $\text{Hom}_D(F(-),Y)$ is a functor $C^{\text{op}}\to \mathsf{Set}$ for fixed $Y$, and $\text{Hom}_D(F(-),-)$ is a functor $C^{\text{op}}\times D\to \mathsf{Set}$.
Ok, now we have a bijection $\alpha(X,Y)\colon \text{Hom}_C(X,G(Y))\to \text{Hom}_D(F(X),Y)$ for all $X$ and $Y$. To say that this family of bijections is natural in $Y$ is to say that for fixed $X$, the family $\alpha(X,-)\colon \text{Hom}_C(X,G(-))\to \text{Hom}_D(F(X),-)$ is a morphism of functors (i.e. it's "functorial"/"natural" in $Y$: lots of "naturality squares" squares commute). Similarly, "natural in $X$" means that for fixed $Y$, the family $\alpha(-,Y)\colon \text{Hom}_C(-,G(Y))\to \text{Hom}_D(F(-),Y)$ is a morphism of functors.
Getting your mind wrapped around all of this takes some doing, and it's best to look at a bunch of examples. This is why I recommended in my comment on your previous question that you pick up an introductory category theory book, which will probably be much easier to learn from.