Proposition 16.2.10 (Functoriality of monoidal Day convolution). Assume the relevant Day convolution monoidal structures exist.
- (1)
-
A lax symmetric monoidal functor \(u\colon C'\to C\) induces a lax symmetric monoidal precomposition functor \[ u^*\colon \Fun (C,D)\longrightarrow \Fun (C',D). \]
- (2)
-
Let \(\Kk \) be the collection of relative slice categories from Proposition 16.2.7. If \(v\colon D\to D'\) is symmetric monoidal and preserves \(\Kk \)-indexed colimits, then postcomposition induces a symmetric monoidal functor \[ v_*\colon \Fun (C,D)\longrightarrow \Fun (C,D'). \]
Proof. Part (1) is induced by precomposition on Day convolution operads. For part (2), postcomposition first gives a lax symmetric monoidal refinement. Since \(v\) is symmetric monoidal and preserves the colimits in the pointwise formulas of Remark 16.2.9, it carries each Day convolution product and the Day convolution unit to the corresponding product and unit in \(\Fun (C,D')\). Its structure maps are therefore isomorphisms. □
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