Is there a reference for the tate construction being lax monoidal?

Question

I know that the functor on spectra sending $X$ to $X^{tC_p}$ is lax monoidal. I am looking for a reference, ideally as early as possible historically, for this fact.

Answer 1

This can be found in the work of Nikolaus--Scholze https://arxiv.org/pdf/1707.01799.pdf. See section 3.