Cartesian closed monoid ($M=\mathrm{End}(D)$)

A monoid encoding a one-object product and exponential structure: pairing is inverse to the map $z\mapsto (\pi z,{\pi }^{\prime }z)$, and currying is inverse to $y\mapsto \epsilon ⟨y\pi ,{\pi }^{\prime }⟩$. These bijections imply pairing naturality $⟨x,y⟩z=⟨xz,yz⟩$ and currying naturality $(x⟨z\pi ,{\pi }^{\prime }⟩{)}^{\ast }=x^{\ast }z$. The element $e=({\pi }^{\prime }{)}^{\ast }$ then satisfies $ez=e$ for every $z$, hence is idempotent. Splitting $e$ supplies a terminal object and a two-object Cartesian closed category whose endomorphisms at the original object are $M$. For the original object to be nonterminal, $M$ must be nontrivial.