Definition. Presheaf [local-0]

Let $A$ be a category. A presheaf over $A$ is a functor of the form

X : A^{op} \to Set

For each object $a \in A$ , we will denote by

X_a := X(a) \in Set

the evaluation of $X$ at $a$ . The set $X_a$ will sometimes be called the fibre of the presheaf $X$ at $a$ , and the elements of $X_a$ thus deserve the name of sections of $X$ over $a$ .

For morphism $a \xrightarrow{u} b$ , the induced map from $X_b \to X_a$ denotes $u^* := X(u)$ .

The category of presheaves over $A$ denotes $\widehat{A}$ .

Definition. Yoneda embedding [local-1]

h : A \to \widehat{A} \\ h_a := \text{Hom}_A(-,a)

Lemma. Yoneda [local-3]

For any presheaf $X$ over $A$ , there is a natural bijection of the form

\begin{aligned} &\theta &&: \text{Hom}_{\widehat{A}}(h_a,X) \xrightarrow{\sim} X_a \\ &\theta(\alpha) &&:= \alpha_a(1_a) \end{aligned}

Proof. [local-2]

We first define inverse map $\tau : X_a \to \text{Hom}_{\widehat{A}}(h_a,X)$ , given a section $s$ of $X$ over $a$ , i.e. $s \in X_a$ , we have

\begin{aligned} \tau(s)_b &: \text{Hom}_A(b,a) \to X_b \\ \tau(s)_b(f) &:= f^*(s) \end{aligned}

for each morphism $b \xrightarrow{f} a$ . This indeed defines a morphism $h_a \xrightarrow{\tau(s)} X$ in $\widehat{A}$ .

Now check $\theta$ and $\tau$ indeed are inverse of each other. First: given $s \in X_a$ , we have

\theta(\tau(s)) = \tau(s)_a(1_a) = 1_a^*(s) = X(1_a)(s) = s

Another direction: given $\alpha : h_a \to X$ , we have

\begin{aligned} \tau(\theta(\alpha))_b(f) &= \tau(\alpha_a(1_a))_b(f) \\ &= f^*(\alpha_a(1_a)) \quad \text{by definition of } \tau \\ &= X(f)(\alpha_a(1_a)) \\ &= \alpha_b(\text{Hom}_A(f, a)(1_a)) \quad \text{by naturality} \\ &= \alpha_b(1_a \circ f) \\ &= \alpha_b(f) \end{aligned}

for each $f : b \to a$ . Naturality is obvious, hence they form a natural bijection.