The category of presheaves are cartesien closed

We will prove the title by using coend calculus, as hinted in Coend Calculus by Fosco Loregian.

For consistence let’s first formulate what to prove:

Theorem. For any (small) category $\mathbf{C}$, its presheaf category $\hat{\mathbf{C}} := [\mathbf{C}^{\mathrm{op}}, {\mathrm{Set}}]$ is cartesien closed, i.e., for any presheaves $P, Q, R \in \hat{\mathbf{C}}$, there exist a exponential object $R^Q$ whi is right adjoint to product:

$$\hat{\mathbf{C}}(P \times Q, R) \cong \hat{\mathbf{C}}(P, R^Q);$$

proof: The exponential object $P^Q$ is defined as $P^Q(c) := \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R)$.

The theorems/propositions in the comments after each equation are all in the book Coend Calculus, and $\mathbf{y}_{()}$ is the standard yoneda embedding.

$$\begin{align} \hat{\mathbf{C}}(P, \hat{\mathbf{C}}(\mathbf{y}_ {()} \times Q, R)) & \cong \int_c {\mathrm{Set}}(Pc, \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R)) &(\text{By Thm 1.4.1}) \\ & \cong \int_c {\mathrm{Set}} (Pc, \int_x {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.4.1}) \\ & \cong \int_c\int_x {\mathrm{Set}} (Pc, {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{Representables preserve limits}) \\ & \cong \int_x\int_c {\mathrm{Set}} (Pc, {\mathrm{Set}}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.3.1}) \\ & \cong \int_x\int_c {\mathrm{Set}} (Pc \times \mathbf{C}(x, c), {\mathrm{Set}} (Qx, Rx)) &(\text{Set is cartesien closed}) \\ & \cong \int_x {\mathrm{Set}} (\int^c Pc \times \mathbf{C}(x, c), {\mathrm{Set}} (Qx, Rx)) &(\text{Representables turn colimts into limits}) \\ & \cong \int_x {\mathrm{Set}} (Px, {\mathrm{Set}} (Qx, Rx)) &(\text{By Prop 2.2.1}) \\ & \cong \int_x {\mathrm{Set}} (Px \times Qx, Rx) &(\text{Set is cartesien closed}) \\ & \cong \hat{\mathbf{C}}(P \times Q, R) &(\text{By Thm 1.4.1}) \\ \end{align}$$

QED.

Source of TeX:

\begin{align}
\hat{\mathbf{C}}(P, \hat{\mathbf{C}}(\mathbf{y}_ {()} \times Q, R)) & \cong \int_c {\rm Set}(Pc, \hat{\mathbf{C}}(\mathbf{y}_c \times Q, R))  &(\text{By Thm 1.4.1}) \\
     & \cong \int_c {\rm Set} (Pc, \int_x {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.4.1}) \\
     & \cong \int_c\int_x {\rm Set} (Pc,  {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{Representables preserve limits}) \\
     & \cong \int_x\int_c {\rm Set} (Pc,  {\rm Set}(\mathbf{C}(x, c) \times Qx, Rx)) &(\text{By Thm 1.3.1}) \\
     & \cong \int_x\int_c {\rm Set} (Pc \times \mathbf{C}(x, c), {\rm Set} (Qx, Rx)) &(\text{Set is cartesien closed}) \\
     & \cong \int_x {\rm Set} (\int^c Pc \times \mathbf{C}(x, c), {\rm Set} (Qx, Rx)) &(\text{Representables turn colimts into limits}) \\
     & \cong \int_x {\rm Set} (Px, {\rm Set} (Qx, Rx)) &(\text{By Prop 2.2.1}) \\
     & \cong \int_x {\rm Set} (Px \times Qx, Rx) &(\text{Set is cartesien closed}) \\
     & \cong \hat{\mathbf{C}}(P \times Q, R) &(\text{By Thm 1.4.1}) \\
        
\end{align}