Definition 4.3.1. Given a category $\mathcal{C}$ the opposite category $\mathcal{C}^{opp}$ is the category with the same objects as $\mathcal{C}$ but all morphisms reversed.
4.3 Opposite Categories and the Yoneda Lemma
In other words $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}^{opp}}(x, y) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, x)$. Composition in $\mathcal{C}^{opp}$ is the same as in $\mathcal{C}$ except backwards: if $\phi : y \to z$ and $\psi : x \to y$ are morphisms in $\mathcal{C}^{opp}$, in other words arrows $z \to y$ and $y \to x$ in $\mathcal{C}$, then $\phi \circ ^{opp} \psi $ is the morphism $x \to z$ of $\mathcal{C}^{opp}$ which corresponds to the composition $z \to y \to x$ in $\mathcal{C}$.
Definition 4.3.2. Let $\mathcal{C}$, $\mathcal{S}$ be categories. A contravariant functor $F$ from $\mathcal{C}$ to $\mathcal{S}$ is a functor $\mathcal{C}^{opp}\to \mathcal{S}$.
Concretely, a contravariant functor $F$ is given by a map $F : \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ and for every morphism $\psi : x \to y$ in $\mathcal{C}$ a morphism $F(\psi ) : F(y) \to F(x)$. These should satisfy the property that, given another morphism $\phi : y \to z$, we have $F(\phi \circ \psi ) = F(\psi ) \circ F(\phi )$ as morphisms $F(z) \to F(x)$. (Note the reverse of order.)
Definition 4.3.3. Let $\mathcal{C}$ be a category.
A presheaf of sets on $\mathcal{C}$ or simply a presheaf is a contravariant functor $F$ from $\mathcal{C}$ to $\textit{Sets}$.
The category of presheaves is denoted $\textit{PSh}(\mathcal{C})$.
Of course the category of presheaves is a proper class.
Example 4.3.4. Functor of points. For any $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there is a contravariant functor
\[ \begin{matrix} h_ U
& :
& \mathcal{C}
& \longrightarrow
& \textit{Sets}
\\ & & X
& \longmapsto
& \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U)
\end{matrix} \]
which takes an object $X$ to the set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U)$. In other words $h_ U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_ U(f) : \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(Y, U)\to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U)$ takes $\phi $ to $\phi \circ f$. We will always denote this presheaf $h_ U : \mathcal{C}^{opp} \to \textit{Sets}$. It is called the representable presheaf associated to $U$. If $\mathcal{C}$ is the category of schemes this functor is sometimes referred to as the functor of points of $U$.
Note that given a morphism $\phi : U \to V$ in $\mathcal{C}$ we get a corresponding natural transformation of functors $h(\phi ) : h_ U \to h_ V$ defined by composing with the morphism $U \to V$. This turns composition of morphisms in $\mathcal{C}$ into composition of transformations of functors. In other words we get a functor
\[ h : \mathcal{C} \longrightarrow \textit{PSh}(\mathcal{C}) \]
Note that the target is a “big” category, see Remark 4.2.2. On the other hand, $h$ is an actual mathematical object (i.e. a set), compare Remark 4.2.11.
Lemma 4.3.5 (Yoneda lemma).reference Let $U, V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Given any morphism of functors $s : h_ U \to h_ V$ there is a unique morphism $\phi : U \to V$ such that $h(\phi ) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\mathcal{C}$ we have a natural bijection
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, F) \longrightarrow F(U), \quad s \longmapsto s_ U(\text{id}_ U). \]
Proof. For the first statement, just take $\phi = s_ U(\text{id}_ U) \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$. For the second statement, given $\xi \in F(U)$ define $s$ by $s_ V : h_ U(V) \to F(V)$ by sending the element $f : V \to U$ of $h_ U(V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)$ to $F(f)(\xi )$. $\square$
Definition 4.3.6. A contravariant functor $F : \mathcal{C}\to \textit{Sets}$ is said to be representable if it is isomorphic to the functor of points $h_ U$ for some object $U$ of $\mathcal{C}$.
Let $\mathcal{C}$ be a category and let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a representable functor. Choose an object $U$ of $\mathcal{C}$ and an isomorphism $s : h_ U \to F$. The Yoneda lemma guarantees that the pair $(U, s)$ is unique up to unique isomorphism. The object $U$ is called an object representing $F$. By the Yoneda lemma the transformation $s$ corresponds to a unique element $\xi \in F(U)$. This element is called the universal object. It has the property that for $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the map
\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U) \longrightarrow F(V),\quad (f : V \to U) \longmapsto F(f)(\xi ) \]
is a bijection. Thus $\xi $ is universal in the sense that every element of $F(V)$ is equal to the image of $\xi $ via $F(f)$ for a unique morphism $f : V \to U$ in $\mathcal{C}$.
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