Proposition 22.5.5. The following statements hold.
- (1)
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If \(C\) is a small symmetric monoidal \(\infty \)-category and \(D\) is presentably symmetric monoidal, then the Day convolution structure on \(\Fun (C,D)\) is presentably symmetric monoidal.
- (2)
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If \(C\) is presentably symmetric monoidal and \(A\in \Alg (C)\), then \(\LMod _A(C)\) and \(\RMod _A(C)\) are presentable. Their forgetful functors to \(C\) create limits and small colimits.
Proof. For (1), the underlying functor category \(\Fun (C,D)\) is presentable by Theorem 22.2.2, and Corollary 16.2.8 equips it with the Day convolution monoidal structure. The category \(D\) is complete because it is presentable, and it is closed by Proposition 22.5.3. Hence Proposition 16.2.13 shows that the Day convolution structure is closed. Its tensor product therefore preserves small colimits separately in both variables.
Part (2) is [Lurie (2017), Corollary 4.2.3.7]. □
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