Symmetric lax monoidal functor takes operads to operads

Question

I am looking for a reference to the following theorem.

If $F:\mathcal{C}\to\mathcal{D}$ is a symmetric lax monoidal functor between symmetric monoidal categories, then it induces a well-defined functor $\underline{F}:\mathrm{Op}_\mathcal{C}\to \mathrm{Op}_\mathcal{D}$ between the categories of operads.

I believe this is a well known fact and I have my own proof of it, but my supervisor suggested me to find a reference instead of writing my own proof if it is already in the literature.

However, I haven't been able to find it so far. I couldn't find it in Loday and Valette's Algebraic Operads nor in Yau's Colored Operads. I also asked some other mathematicians and they didn't know about a reference for this, even though they agree it is well known.

Can anyone give me a reference for the above result?

Answer 1

It's in Fresse's book Homotopy of Operads and Grothendieck-Teichmüller Groups, volume I, Proposition 3.1.1(a).