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Let $X$ be a scheme. Let $S$ be the site of open subschemes with the Zariski topology, and let $S'$ be the site of open affine subschemes with the Zariski topology. Let $T$ and $T'$ be the associated toposes. Let $f\colon T\to T'$ be the topos map where $f^*(U)=U$ for any affine open subscheme $U$. Then $f$ is an equivalence because open affines form a base for the topology. Let $g$ be its inverse. Then $g^*$ does not restrict to a map of sites: If $V$ is an open subscheme, then $g^*(V)$ is the sheaf "represented by $V$" (i.e. it sends an affine open $U$ to $\mathrm{Hom}_X(U,V)$), but since if $V$ is not affine, then $g^*(V)$ is not represented by an object in $S'$.
Let $X$ be a scheme. Let $S$ be the site of open subschemes with the Zariski topology, and let $S'$ be the site of open affine subschemes with the Zariski topology. Let $T$ and $T'$ be the associated toposes. Let $f\colon T\to T'$ be the topos map where $f^*(U)=U$ for any affine open subscheme $U$. Then $f$ is an equivalence because open affines form a base for the topology. Let $g$ be its inverse. Then $g^*$ does not restrict to a map of sites: If $V$ is an open subscheme, then $g^*(V)$ is the sheaf "represented by $V$" (i.e. it sends an affine open $U$ to $\mathrm{Hom}_X(U,V)$), but since $V$ is not affine $g^*(V)$ is not represented by an object in $S'$.