# Functoriality of infinite loop space machines?

If $$C$$ is a symmetric monoidal category, then $$BC$$ is canonically an algebra over a certain $$E_\infty$$ operad, but if $$F: C \to D$$ is a symmetric monoidal functor then (as far as I can see) $$BF: BC \to BD$$ is not a map of algebras over that operad (unless all the morphisms $$(Fx) \otimes (Fy) \to F(x \otimes y)$$ are identities).

Because of this I struggle to associate a spectrum map between the two spectra arising from $$BC$$ and $$BD$$ by infinite loop space theory, I can only see how to get a zig-zag where the wrong-way map is a weak equivalence. For most practical purposes that's just as good, but nevertheless I wonder: are any of the "well known" infinite loop space machines functorial on the nose, with respect to symmetric monoidal functors?

Here is a nice gentle old-fashioned answer. Symmetric monoidal categories are functorially equivalent as symmetric monoidal categories to permutative (symmetric strict monoidal) categories, and those are functorially equivalent (essentially the same as) algebras over a certain $$E_{\infty}$$ operad $$\mathcal{P}$$, known nowadays as the categorical Barratt-Eccles operad, in Cat. Since $$B$$ is product preserving it gives a functor from $$\mathcal P$$-algebras in Cat to $$B\mathcal P$$-algebras in Top. That gives $$B$$ as a functor from symmetric monoidal categories to algebras over an $$E_\infty$$ operad. That goes back, at least in outline, to my 1974 paper $$E_{\infty}$$ spaces, group completions, and permutative categories".
Again in outline, by two recent papers, the same argument works equivariantly for (genuine) symmetric monoidal $$G$$-categories, which give genuine $$G$$-spectra for finite groups $$G$$ via infinite loop $$G$$-space machines. See Equivariant iterated loop space theory and permutative $$G$$-categories http://www.math.uchicago.edu/~may/PAPERS/GM3.pdf and Symmetric monoidal $$G$$-categories and their strictification http://www.math.uchicago.edu/~may/PAPERS/AddCat1.pdf}.