= Solution
For a small category $C$, write $\widehat C=[C^{\mathrm{op}},\mathbf{Set}]$. Its objects are <categorical presheaves> and its arrows <natural transformations>. The terminal presheaf is the constant singleton, and products and <equalizers> are computed objectwise. Their restriction maps are induced by the corresponding set operations, so the objectwise universal properties are natural and give <finite limits> in $\widehat C$.
For presheaves $F,G$, define the <exponential object> by
$$
(G^F)(A)=\operatorname{Nat}(yA\times F,G),\qquad yA=C(-,A).
$$
For $u:B\to A$, restriction precomposes with $yu\times1_F$. Evaluation sends $(\alpha,x)$ at $A$ to $\alpha_A(1_A,x)$. Given $h:H\times F\to G$, its transpose sends $z\in H(A)$ to the transformation whose component at $B$ is
$$
(v:B\to A,x\in F(B))\longmapsto h_B(H(v)z,x).
$$
Naturality of $h$ proves this is a <natural transformation> and that the assignment is natural in $A$. Conversely evaluation reconstructs $h$. These constructions are inverse, proving the exponential universal property explicitly.
For the <subobject classifier>, let $\Omega(A)$ be the set of <sieves on a category> on $A$ and let $\Omega(u)$ take inverse image of <sieves on a category>. The truth map $1\to\Omega$ selects the maximal <sieve on a category> at every stage. A <monomorphism> of presheaves is objectwise injective, so identify it with $U\subseteq F$. Define
$$
\chi_U(x)=\{v:B\to A:F(v)x\in U(B)\},\qquad x\in F(A).
$$
This is a <sieve on a category>, and restriction gives precisely inverse image, so $\chi_U$ is natural. Its value is maximal exactly when $x\in U(A)$, by testing the identity arrow. Thus $U$ is its pullback of truth. Uniqueness follows since membership of $v$ is detected by whether the restriction to $v$ is true. Smallness of $C$ makes all the displayed components and natural-transformation sets genuine sets. Hence \b[$\widehat C$ is an elementary topos].
The same constructions work componentwise under the weaker slice hypothesis. Indeed,
$$
\operatorname{Nat}(yA\times F,G)\cong\operatorname{Nat}(F\circ\mathrm{dom},G\circ\mathrm{dom})_{C/A}.
$$
A component on $v:B\to A$ is simply a map $F(B)\to G(B)$, with the naturality conditions for triangles in $C/A$. Restricting to a small skeleton of an essentially small slice therefore makes this a set; the result determines the maps on all isomorphic slice objects. Likewise a <sieve on a category> on $A$ is a full downward-closed collection of objects of $C/A$, determined by its intersection with that small skeleton, so the collection of <sieves on a category> is a set. The restriction, evaluation and classifier proofs above are unchanged. This establishes the <slice-small presheaf construction>.
For an example, take a discrete category with one object for every ordinal and no arrows other than identities. Each slice is the terminal category, but there is a proper class of pairwise nonisomorphic objects, so it is not equivalent to a small category.
\b[The large-category assertion needs a universe convention.] The constructions give all elementary-topos axioms in a larger universe in which those diagrams and their transformations are admitted. They do not imply local smallness in the original universe: for the discrete example, transformations from the constant singleton to the constant two-element presheaf are arbitrary ordinal-indexed binary families and do not form a set there. If “topos” is required to be locally small relative to that original universe, the unrestricted large-category assertion is false. With the usual larger-universe interpretation, the explicit component constructions prove the intended result; no assumption that the original category itself is essentially small is needed.
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