A presheaf category on a small category is cartesian closed

By [L4], binary products in $\widehat{\mathcal{C}}$ are computed pointwise, so $H\times P$ is the presheaf with $(H\times P)(c)=H(c)\times P(c)$. The assignment $c\mapsto \mathrm{Nat}(y(c)\times P,Q)$ with restriction by precomposition along $y(f)\times 1_P$ is therefore a presheaf $Q^P$.

Given $\alpha :H\to Q^P$, define $\tilde{\alpha }:H\times P\to Q$ by sending $(h,p)\in H(c)\times P(c)$ to the element of $Q(c)$ corresponding under [L2] to the natural transformation $y(c)\stackrel{h}{\to }H\stackrel{\alpha }{\to }{Q}^P$, evaluated at $p$. Conversely, given $\beta :H\times P\to Q$, use [L2] and the set-level currying of [L3] objectwise to define ${\widehat{\beta }}_c(h)\in \mathrm{Nat}(y(c)\times P,Q)$; naturality in $c$ is exactly the restriction rule from step 1.1.

The two constructions of step 2.1 are inverse because Yoneda identifies a natural transformation out of $y(c)$ with its value at $1_c$, and the set-level currying and uncurrying in [L3] are inverse. Hence $\widehat{\mathcal{C}}(H\times P,Q)\cong \widehat{\mathcal{C}}(H,Q^P)$ naturally in $H$.

Step 3.1 shows that $-\times P$ has right adjoint $(-{)}^P$, while [L4] gives the finite products. Therefore [L5] implies that $\widehat{\mathcal{C}}$ is cartesian closed.