Corollary 14.4.11. Let \(C\) be a monoidal \(\infty \)-category with a terminal object \(T\). Then \(T\) carries a unique associative algebra structure, and it is terminal in \(\Alg (C)\). When \(C\) is symmetric monoidal, the same holds for the commutative algebra structure and \(\CAlg (C)\).
Proof. Apply Corollary 14.4.10 to the empty diagram. It produces a terminal algebra \(A\) whose underlying object is \(T\). If \(B\) is any other algebra with underlying object \(T\), the unique map \(B\to A\) has an isomorphism as its underlying morphism, and is therefore itself an isomorphism. Moreover, both \(\Hom _{\Alg (C)}(B,A)\) and \(\Hom _C(T,T)\) are contractible, so the anima of such maps lying over \(\id _T\) is contractible. It follows that the fiber of algebra structures on \(T\) is contractible. The commutative case is identical. □
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