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.
For the second part of the question, a natural transformation $\epsilon : D \to D'$ defines a cocone on $D$ after composing with the colimit cocone on $D'$. Indeed a cocone on $D$ is a natural transformation from the functor $D$ to a constant functor on $\mathbf I$. The colimit cocone on $D'$, let us call it $\gamma$, is a natural transformation from $D'$ to the constant functor having values the colimit of the diagram $D'$. Hence $\gamma \circ \epsilon$ is a natural transformation from $D$ to the constant functor having value the colimit of $D'$. By universal property of the colimit cocone on $D$, we get a map from the colimit of $D$ to the colimit of $D'$.
In your drawing, the purple arrows are $\gamma \circ \epsilon$, you just need to add the arrows from $\gamma_i : D'(i) \to \mathrm{colim}D'$.
So now we have a map $\mathrm{colim}(\epsilon) : \mathrm{colim}D \to \mathrm{colim} D'$, this induces a natural transformation $\mathscr{A}(\mathrm{colim} D',-) \to \mathscr{A}(\mathrm{colim}D,-)$, often called $\mathrm{colim}(\epsilon)^*$.
This is just the general fact that $f : A \to A'$ induces a the natural transformation of covariant functors on $\mathscr{A}$, $f^* : \mathscr{A}(A',-) \to \mathscr{A}(A,-)$, (recall the (co)yoneda embedding $\mathscr{A}^{op} \to \mathbf{Fun}(\mathscr{A},\mathsf{Set})$).
The first variable in the hom functors is contravariant !
Best Answer
You are on the right track. The diagonal functor is defined by $\ \Delta: C\to C\times C$. On objects: $c\mapsto c\times c$. On arrows: $(f,f):(c, c)\to (c', c')$.
From this, we can construct a right adjoint simply by enforcing the rules:
A right adjoint is a functor $G:C\times C\to C$ such that
$\text{Hom}_C(c,G(a,b))\cong \text{Hom}_{C\times C}(\Delta c,(a,b)).$
The left-hand side of this is clear. The right-hand side is a set of arrows of the form $(c,c)\to (a,b)$ in the category $C\times C.$ But these arrows, by definition, are $pairs\ (f,g)$ where $f:c\to a$ and $g:c\to b$, so there is the obvious bijection $\text{Hom}_{C\times C}(\Delta c,(a,b))\cong \text{Hom}_{C}( c,a)\times \text{Hom}_{C}(c,b)$. This is natural as well, but that's not important here. It just gives us an idea of what $G should $ be.
That is, $G=\times$, the product functor. On objects: $(a,b)\to a\times b$ and on arrows, the unique morphism induced by the UMP of the product.
As long as $C$ has products and small hom-sets, this will work.
The obvious unit is $\eta_c=\langle 1_c,1_c\rangle .$
All that remains is to check that $G\Delta f\circ \eta_c=f$ for $f:c\to a\times b.$
I guess you can take it from here.