The well-known duality principle asserts that if a statement is true (resp. false) for a category $\mathfrak{C}$, then the dual statement must be also true (resp. false) for the dual category $\mathfrak{C}^{\mathrm{op}}$. Now, in some cases it is useful to work with the dual category because it might simplify the analysis. However, in this case, by the duality principle, we translate the found properties with respect to $\mathfrak{C}$. Hence, my quedtion is: except for what I remarked before, why it is so important to study dual categories? Honestly, I considered such a notion useful only to simplify the work needed when using $\mathfrak{C}$.
Duality in category theory
category-theory
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The most important cofibrantly generated model categories are the combinatorial ones, which are also locally presentable; by Dugger’s theorem these are all Quillen equivalent to left Bousfield localizations of the projective model structure on some category of simplicial presheaves. Such categories are a lot like the classical model category of simplicial sets: you build things out of cell complexes, most notably.
Certainly the opposite of a cofibrantly generated model category is fibrantly generated, but this corresponds roughly to taking presheaves valued in the opposite category of simplicial sets, which is not as important. This is a homotopy theory analogue of the fact that locally presentable categories are more practically important than their opposites, though the notions are formally equivalent by duality. A particular problem with getting fibrant generation is the shortage of cosmall objects in most “natural” categories.
The idea of duality is that, whenever you have a purely categorical statement (or concept), you can look at what that statement means for a category $ C $ when applied to $ C^{\mathrm{op}} $: this gives you the dual statement or concept. Then, if you have a statement that is true in any category, then its dual statement is also necessarily true, since it is true for $ C^\mathrm{op} $. Finding out the dualized statement is then just a matter of properly writing down the statement in the dual category and interpreting it in the original one (which can often be tedious).
As an example, you can see how the definition of an initial object in $ C^{\mathrm{op}} $ coincides with that of a terminal object in $ C $! Thus, the dualized statement of "all initial items are uniquely isomorphic" is "all terminal objects are uniquely isomorphic". Same goes for limits/colimits, continuous/co-continuous functors, left/right-adjoints (note here that there are two categories involved, both of which you end up dualizing), left/right Kan extensions, kernels/cokernels, images/coimages in an abelian category, etc. In the end, you can apply duality to whatever statement you want, it is just a matter of practicing that conceptually simple but powerful tool.
I do think though that the lack of details is just typical of users of categories in general, and not just of duality, but that is another topic altogether.
Best Answer
What is important is that dual categories appear naturally, most notably because there are (what people call) "contravariant" functors - what I prefer to call functors on dual categories (it is very convenient to declare all functors as "covariant"). A basic example is the functor $\mathbf{Top}^{\mathrm{op}} \to \mathbf{CRing}$, $X \mapsto C(X)$, $f \mapsto f^*$, a more complicated example is singular cohomology $H^* : \mathbf{Top}^{\mathrm{op}} \to \mathbf{grAb}$. Also, every (pre)sheaf on a space $(X,\mathcal{O})$ is actually a functor on $\mathcal{O}^{\mathrm{op}}$.
One important observation is also that dual categories of concrete categories in practice behave quite differently than these concrete categories. For example, although the dual of every abelian category is also abelian, the dual of a Grothendieck category is almost never a Grothendieck category, which applies in particular to $(\mathbf{Mod}_R)^{\mathrm{op}}$, the dual of the category of right $R$-modules, where $R$ is a nontrivial ring.
Also, concrete representations of these dual categories often involve some kind of topology. In fact, Pontryagin duality tells us (in a special case) that there is an equivalence of categories between $\mathbf{Ab}^{\mathrm{op}}$ and the category $\mathbf{CompAb}$ of compact topological abelian groups. It follows formally that $(\mathbf{Mod}_R)^{\mathrm{op}}$ is equivalent to the category ${}_R \mathbf{CompMod}$ of compact topological left $R$-modules.