= Solution
Use the <elementary topos> definition: a <category> with <finite limits>, <exponential objects>, and a <subobject classifier>. The indexing category $\mathbb C$ is assumed small, as usual, and $\widehat{\mathbb C}=[\mathbb C^{\mathrm{op}},\mathbf{Set}]$ denotes its <presheaf category>. A <finite limit> of <categorical presheaves> is formed objectwise: at $c$ take the corresponding limit of sets, and apply the universal property to induce restriction maps. The terminal <categorical presheaf> is the constant singleton, so this supplies all required <finite limits>.
For <categorical presheaves> $A,B$, write $yc=\mathbb C(-,c)$ and define
$$
(B^A)(c)=\operatorname{Nat}(yc\times A,B).
$$
For $h:d\to c$, restriction is precomposition with $yh\times1_A$. Evaluation sends $(\theta,a)$ at $c$ to $\theta_c(1_c,a)$. Given $\phi:H\times A\to B$, its transpose sends $x\in H(c)$ to the <natural transformation> whose component at $d$ is
$$
(h:d\to c,a\in A(d))\longmapsto\phi_d(H(h)x,a).
$$
Naturality of $\phi$ proves this is a <natural transformation>; naturality in $c$ proves it is a map $H\to B^A$. Conversely evaluation recovers $\phi$, and these constructions are inverse. Therefore
$$
\widehat{\mathbb C}(H\times A,B)\cong\widehat{\mathbb C}(H,B^A),
$$
so the <presheaf category> is <Cartesian closed>.
Define $\Omega(c)$ to be the set of <sieves on a category> with codomain $c$. For $h:d\to c$, let $\Omega(h)S=h^*S=\{k:e\to d:hk\in S\}$, and let $\top:1\to\Omega$ select the maximal sieve at each object. For a <subobject> $U\hookrightarrow X$, define its characteristic map by
$$
\chi_c(x)=\{h:d\to c:X(h)x\in U(d)\}.
$$
The restriction closure of $U$ makes this a <sieve on a category>, and the formula commutes with pullback of <sieves on a category>, so $\chi$ is natural. It is maximal exactly when $1_c$ belongs to it, equivalently when $x\in U(c)$; hence the pullback of $\top$ along $\chi$ is $U$. Uniqueness is also explicit: naturality forces $h\in\chi_c(x)$ exactly when the restriction $X(h)x$ lies in the truth pullback at $d$. \b[Thus the sieve presheaf is a subobject classifier, and every small presheaf category is an elementary topos.]
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