Lemma 16.4.3 (Bilinear maps). Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. For \(A,B,M\in \CMon (C)\) there is a natural equivalence \[ \Hom _{\CMon (C)}(A\otimes B,M) \simeq \Nat _{\Span (\Fin )^2} \bigl (A(-)\otimes _C B(-),M(-\times -)\bigr ). \] The same formula holds in \(\CGrp (C)\) for commutative group objects. We refer to objects of the anima on the right as bilinear maps from \(A\) and \(B\) to \(M\).
Proof. The tensor product \(A\otimes B\) is the reflection of the ambient Day convolution \(A\otimes _{\Day }B\). Since \(M\) is local, maps from this reflection to \(M\) are the same as maps from \(A\otimes _{\Day }B\) to \(M\). The universal property of Day convolution, in the form of Proposition 16.2.6, identifies the latter anima with the displayed anima of natural transformations. The argument for commutative groups is identical. □
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