Why do we need to use opposite categories/contravariant functors I know this is obviously a really dumb question apologies, but I’m currently trying to learn Category Theory and I’m struggling to find a resource/textbook that doesn’t just define these concepts in a really unmotivated way and then move on.
Looking at an opposite category doesn’t seem to add any new or interesting structure to the things you’re studying, and the correspondence between morphisms is so natural and trivial that it seems really unclear what you can actually “do” with the opposite category that you can’t just do with C. Contravariant functors then just seem to be less intuitive ways to talk about basically the same stuff
Clearly this is just me being dumb/not finding the right material yet but can someone point out what I’m missing? Even just an example where thinking in terms of opposite categories/contravariant functors is actually much more natural/illuminating?
 A: The natural starting point is that there are many examples of contravariant operations "in nature." Here are a few examples of many. It suffices to understand and appreciate the first.

*

*For instance, given a map of vector spaces (over a fixed ground field $k$), $f:V\to W$, there is an induced map of the dual spaces $f^*:W^*\to V^*$ given by $f^*(\phi)=\phi\circ f$. Moreover, given maps
$$
V\xrightarrow{f}W\xrightarrow{g} X
$$
we have
$$
X^*\xrightarrow{g^*}W^*\xrightarrow{f^*}V^*
$$
such that $(g\circ f)^*=f^*\circ g^*$. Of course, $\Bbb{1}_V^*=\Bbb{1}_{V^*}$. These are the conditions of dualization $V\mapsto V^*$ being a contravariant functor from the category of vector spaces to itself.


*Let $\mathcal{A}$ be some reasonable algebraic category (though it need not be really) like the category of groups, rings, etc. Then for a fixed object $A$, the functor $\mathrm{Hom}(-,A):\mathcal{A}\to \rm{Sets}$ given by sending $B\mapsto \mathrm{Hom}(B,A)$ is a contravariant functor. Taking $\mathcal{A}$ to be the category of vector spaces over $k$ and $A=k$ we recover the previous example.


*In algebraic geometry, one associates to an affine variety $X\subseteq \Bbb{A}^n_k$ its coordinate ring $A(X)$ which is a $k-$algebra of finite type. Given a morphism of varieties $\phi:X\to Y$, we get a map the other way of coordinate rings: $\phi^*:A(Y)\to A(X)$. This defines a contravariant functor from the category of affine varieties to the category of $k-$algebras.


*To a manifold $M$, we can associate $\Omega^*(M)$ - the (differential graded) algebra of differential forms on $M$. A map of manifolds $f:M\to N$ induces a map $f^*:\Omega^*(N)\to \Omega^*(M)$ and one proves that this defines a contravariant functor from the category of smooth manifolds to the category of (differential graded) algebras.


*We can define a functor $\mathrm{Bun}(-)$ which assigns to a topological space $X$ the set $\mathrm{Bun}(X)$ of vector bundles over $X$. Given a map of topological spaces $f:X\to Y$, a vector bundle $E\to Y$ pulls back to a vector bundle $f^*E\to X$. In particular, we get an induced map $f^*:\mathrm{Bun}(Y)\to \mathrm{Bun}(X)$. This defines a contravariant functor from topological spaces to sets.


*In topology one studies cohomology functors which assign to a topological space $X$ a sequence of groups $H^i(X)$ for $i\ge 0$. A map of spaces $f:X\to Y$ turns out to induce a map $f^*:H^i(Y)\to H^i(X)$ for each $i\ge 0$. These are contravariant functors from topological spaces to Abelian groups.
Once one has realized that objects behaving like contravariant functors occur in abundance, it is useful to give such objects a name. This is all that we are doing with the definition of contravariant functor. The opposite category is important as it shows that all functors can be understood as covariant functors. Indeed, it reduces the study of contravariant functors from a category $\mathcal{C}$ to the study of covariant functors on $\mathcal{C}^{\mathrm{op}}$. In short, this tells us that as far as category theory is concerned one can basically always speak of "functors" without particular concern as to whether one means covariant or contravariant when it comes to stating theorems.
A: Here is another example of a beautiful duality:
Given a compact Hausdorff space $X$, we can look at the commutative, unital $C^*$-algebra $$C(X) = \{f: X \to \mathbb{C}: f \mathrm{ \ is \ continuous}\}.$$
Given a continuous map $f: X \to Y$ between compact Hausdorff spaces, the mapping
$$C(f): C(Y) \to C(X): g \mapsto g \circ f$$
gives us a $*$-homomorphism. In this way, we obtain the contravariant functor
$$C(-): \operatorname{CHaus}\to \operatorname{C^*}$$
where $\operatorname{C^*}$ denotes the category of unital, commutative $C^*$-algebras. In fact, this functor defines an (anti)equivalence of categories.
This functor motivates the idea of $C^*$-algebra theory as being non-commutative topology. A modification of this idea leads to an anti-equivalence between the categories of compact Hausdorff topological groups and the notion of compact quantum group (as introduced by Woronowicz in the nineties).
