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Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in each variable. (What I'm about to say is true in greater generality, but let's just stick with these assumptions.)

Then the category $\mathrm{Fun}(A,C)$ of functors $A\to C$ can be endowed with the Day convolution product, which is a nice symmetric monoidal structure [LNM 137, pp. 1-38]. Here's the fact I'm interested in:

Proposition. There is a natural equivalence of categories between the category of lax (symmetric) monoidal functors and the category of (commutative) monoids in $\mathrm{Fun}(A,C)$ with respect to the Day convolution.

After a fair amount of searching, the earliest reference I've found for such a result is Mandell-May-Schwede-Shipley, Model categories of diagram spectra, Pr. 22.1.

But surely it goes much further back than that. In particular, I'd imagine that Brian Day himself would have known this in the 70s. It seems like the sort of thing that's actually a special case of something incredibly general that category theorists knew 40 years ago. So ...

Question. Where is the first place this result (or a generalization thereof) appears in the literature?

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up vote 7 down vote accepted

This observation appears already in Day's thesis as Example 3.2.2. For some reason it is only stated for commutative monoids and symmetric (pro)monoidal functors and only as a correspondence of objects not an equivalence of categories, also no proof is given. (Day's thesis used to be available on Street's homepage but the link is dead now.)

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This is really great. I'm gratified to know that, indeed, Day knew this fact from the beginning. Thank you, Karol. –  Clark Barwick May 15 '13 at 11:15
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