Exponential in a presheaf category ($(Q^P)(c)=\mathrm{Nat}(H_c\times P,Q)$)

For categorical presheaves $P,Q$ on a small category, restrictions in the displayed exponential object act by precomposition with $H_u\times 1_P$. Evaluation sends $(\alpha ,x)$ at $c$ to ${\alpha }_c(1_c,x)$. The currying of $t:R\times P\to Q$ sends $z\in R(c)$ to the natural transformation whose value on $(v:a\to c,x\in P(a))$ is $t_a(R(v)z,x)$. Evaluation and currying are inverse and natural, proving the Cartesian closed category structure.