Topos Theory 的 1.12

令 $X, Y, Z \in \widehat{C}$ 為 category $C$ 的 presheaves，那麼

\text{hom}(Z, Y^X) \cong \text{hom}(Z \times X, Y)

Proof. [local-0]

\begin{aligned} \text{hom}(Z, Y^X) &\cong \text{hom}(\text{colim}(h_\alpha), Y^X) \quad (Z \text{ 可以表示為 representables 的 colimit}) \\ &\cong \lim \text{hom}(h_\alpha, Y^X) \quad (\text{contravariant}) \\ &\cong \lim \text{hom}(h_\alpha \times X, Y) \quad (Y^X \text{的定義}) \\ &\cong \text{hom}(\text{colim}(h_\alpha \times X), Y) \quad (\text{contravariant}) \\ &\cong \text{hom}(\text{colim}(h_\alpha) \times X, Y) \quad ((-)\times X \text{ preserves colimits in } \text{Sets} \text{, hence in } \widehat{C}) \\ &\cong \text{hom}(Z \times X, Y) \end{aligned}