= Solution
For a <locally small category> $\mathcal C$, the covariant <Yoneda lemma> gives a bijection
$$
\boxed{\operatorname{Nat}(\mathcal C(c,-),H)\cong H(c),\qquad\tau\longmapsto\tau_c(1_c),}
$$
natural in $c$ and the <functor> $H:\mathcal C\to\mathbf{Set}$. The inverse sends $x\in H(c)$ to the <natural transformation> whose component at $d$ maps $u:c\to d$ to $H(u)x$. Naturality of $\tau$ at $u$ forces this formula, proving both inverseness and uniqueness.
Write $\mathcal E=\int F$ for the <category of elements> of $F$. Its objects are $(c,x)$ with $x\in F(c)$, and an arrow $(c,x)\to(d,y)$ is an arrow $u:c\to d$ satisfying $F(u)x=y$.
Given an object $\alpha:P\to F$ of the <slice category> $[\mathcal C,\mathbf{Set}]/F$, define
$$
\Phi(\alpha)(c,x)=\{p\in P(c):\alpha_c(p)=x\}.
$$
An arrow $u:(c,x)\to(d,y)$ acts by the restriction of $P(u)$; <naturality> of $\alpha$ makes this restriction well-defined. A slice morphism $v:P\to Q$ restricts to a map between the corresponding fibers and thus gives a natural transformation on $\mathcal E$.
Conversely, given $H:\mathcal E\to\mathbf{Set}$, set
$$
P_H(c)=\coprod_{x\in F(c)}H(c,x),\qquad P_H(u)(x,h)=(F(u)x,H(u)h),
$$
and let $\alpha_H:P_H\to F$ be projection to $x$. The <functor> laws and naturality follow from those for $F$ and $H$. On morphisms use the maps on each summand. Taking a fiber of this disjoint union recovers $H(c,x)$ canonically. Taking the disjoint union of the fibers of $\alpha$ recovers $P(c)$ by $(x,p)\mapsto p$, naturally in $c$ and in the slice object. These are inverse <natural isomorphisms>, proving
$$
\boxed{[\mathcal C,\mathbf{Set}]/F\simeq[\mathcal E,\mathbf{Set}].}
$$
This is the <slice of a set-valued functor category> construction.
Now assume $\mathcal C$ is a <small category>, so $\mathcal E$ is small. Define a <diagram in a category>
$$
D:\mathcal E^{\mathrm{op}}\longrightarrow[\mathcal C,\mathbf{Set}],\qquad D(c,x)=\mathcal C(c,-).
$$
For $u:(c,x)\to(d,y)$ in $\mathcal E$, its reverse arrow acts by precomposition $\mathcal C(d,-)\to\mathcal C(c,-)$. There is a <cocone> to $F$ with component $g\mapsto F(g)x$. To verify its colimit property directly, use pointwise <colimits> of sets. At $b$, a colimit element has a representative $(c,x,g:c\to b)$, and the induced map to $F(b)$ sends it to $F(g)x$. This map is surjective, since $z\in F(b)$ is represented by $(b,z,1_b)$. Moreover, the element-category arrow $g:(c,x)\to(b,F(g)x)$ gives precisely the colimit relation
$$
[(c,x,g)]=[(b,F(g)x,1_b)].
$$
Thus all representatives with the same image are equal in the colimit, proving injectivity. The resulting bijections are natural in $b$. We have proved the <canonical colimit presentation of a covariant set-valued functor>:
$$
\boxed{F\cong\operatorname{colim}_{(c,x)\in\mathcal E^{\mathrm{op}}}\mathcal C(c,-).}
$$
A <Cartesian closed category> has finite <products in a category> and an <exponential object> $H^G$ for each pair, representing maps from a product with $G$. In $[\mathcal C,\mathbf{Set}]$, the terminal functor and binary products are computed pointwise. Define the <exponential of covariant set-valued functors> by
$$
(H^G)(c)=\operatorname{Nat}(\mathcal C(c,-)\times G,H).
$$
Smallness makes this a set. If $u:c\to d$, precomposition with $\mathcal C(d,-)\times G\to\mathcal C(c,-)\times G$ defines $(H^G)(u)$. Functoriality is immediate from associativity of composition. Its <evaluation map of an exponential object> is
$$
\operatorname{ev}_c(\tau,z)=\tau_c(1_c,z).
$$
For $u:c\to d$, naturality of $\tau$ shows $H(u)\tau_c(1_c,z)=\tau_d(u,G(u)z)$, which is exactly the naturality equation for evaluation.
Given $\theta:P\times G\to H$, define its <currying> at $p\in P(c)$ by
$$
(\widehat\theta_c(p))_b(f,z)=\theta_b(P(f)p,z),\qquad f:c\to b,\quad z\in G(b).
$$
Naturality of $\theta$ proves that this is a natural transformation in $b$, and its compatibility with arrows $c\to d$ proves that $\widehat\theta:P\to H^G$ is natural. Evaluation recovers $\theta$ by setting $f=1_c$. Conversely, for $\sigma:P\to H^G$, its naturality implies
$$
(\sigma_c(p))_b(f,z)=(\sigma_b(P(f)p))_b(1_b,z),
$$
so currying the evaluation composite recovers $\sigma$. These formulas give the required <natural bijection>, hence \b[$[\mathcal C,\mathbf{Set}]$ is Cartesian closed].
Finally, for every $F$ the small category $\mathcal E=\int F$ has a Cartesian closed functor category by the same construction. The slice equivalence above transports its terminal object, products and exponentials to $[\mathcal C,\mathbf{Set}]/F$. The original functor category also has pointwise finite limits. Therefore \b[it is locally Cartesian closed]. Every construction here is explicit; no adjoint functor theorem is used.
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