How is the notion of adjunction of two functors usefull? Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ?
Thank you in advance for your help.
 A: It's best to look at some of the most obvious examples of adjunctions to get a feel for what they do. For example, in any algebraic category (groups, vector spaces, etc.) a left adjoint to the forgetful functor is a free functor. This gives a formal definition for what it means for an object to be "free."
Another really prevalent case is that of limits (or colimits). If a (co)limit exists for all diagrams of a given shape then this defines an adjunction. Limits and colimits are prevalent in every area of mathematics, so from these alone you can see that adjunctions are everywhere.
Another good example is a tensor-hom adjunction. This formalizes the intuitive idea of an "object of mappings" between any two objects. In a monoidal category, if you have a right adjoint to the functor $(-)\otimes X$ for any object $X$ then this is called the internal hom. If such a right adjoint exists, the category is said to be closed monoidal. 
This is an extremely awesome property for a category to have, mainly because it allows you to curry arrows out of monoidal products. For example, in the category of commutative ring modules, the obvous internal hom corresponds to the tensor product. In the category of sets, it is the usual function sets. If you work in a convenient category of topological spaces (like k-spaces) which is closed with respect to the Cartesian product, then you can say things like "a homotopy $H$ between two paths $a$, $b$ in a space $X$ is a path in the space of paths $X^{[0,1]}$ joining $a$ to $b$."
A: I have at least two answers for this.
Every adjunction prepares an equivalence of categories.
Namely, if $(F,G,\eta,\varepsilon) : \mathcal{C} \to \mathcal{D}$ is an adjunction, let $\mathsf{Fix}(\eta) \subseteq \mathcal{C}$ denote the full subcategory of objects $X \in \mathcal{C}$ such that $\eta(X) : X \to G(F(X))$ is an isomorphism. We define $\mathsf{Fix}(\varepsilon) \subseteq \mathcal{D}$ analogously. Then, $F,G$ restrict to an equivalence of categories
$$\mathsf{Fix}(\eta) \simeq \mathsf{Fix}(\varepsilon).$$
This is also useful in the case of preorders, where adjunctions are known as Galois connections. Many famous equivalences of categories arise this way (and this seems to be kept as a secret).


*

*The main theorem of Galois theory

*Hilbert's Nullstellensatz

*The theorem of Gelfand-Naimark

*The duality between affine schemes and commutative rings

*The duality of finite-dimensional vector spaces

*Pontrjagin duality

*Grothendieck's main theorem Galois theory

*...


Of course, in all these theorems, the hard part is to determine the fixed objects. But the framework for these theorems is provided by basic adjunctions.
Left adjoints are cocontinuous.
That is, if $F$ is left adjoint to a functor $G$, then $F$ preserves colimits. This is a standard fact, and it is easy to prove, but it subsumes hundreds of concrete isomorphisms which you have (or will) come across. For example, $$S^{-1}(\bigoplus_i M_i) \cong \bigoplus_i S^{-1} M_i$$ (localization of modules) is a formal consequence, as is $$(G * H)^{\mathrm{ab}} \cong G^{\mathrm{ab}} \oplus H^{\mathrm{ab}}$$ (abelianization of groups). We don't have to compute anything to get these isomorphisms.
The tensor product is right exact since tensoring with some module has a right adjoint, the (internal) hom functor. We don't have to fiddle around with cokernels, which is done in many texts which avoid adjoints and prove right exactness directly.
There is even a sort-of converse of this general result, namely Freyd's general adjoint functor theorem: Every cocontinuous functor, which has a solution set (which is a quite mild set-theoretic condition), is a left adjoint. So the preservation of colimits by left adjoints (and, dually, of limits by right adjoints) is perhaps one of their main goals.
