In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the question, which is about the converse, in my words is the following

Given a functor between monoidal categories that preserves monoids and morphisms of monoids on these categories, does it follow that the functor is lax monoidal?

I am interested in this question because in my research I've encountered a functor between the underlying collections of operads that do preserve the operad structure as well as the morphisms of operads. Since operads are monoids on the category of collections (or $\mathbb{S}$-modules for the symmetric case) with the composite of collections as monoidal product, I was expecting this functor to be lax monoidal, but I haven't been able to prove it.

If the answer to my above question is positive it must mean that I have made some mistake and I would post a question related to that particular case. If not, I am fine with my functor preserving just operads and its morphisms. In fact, I believe that this implication is not going to be true in general, because the associativy axiom of monoidal categories relates 3 (in general different) objects while the axioms of monoids only involve a single object. But maybe there are some nice sufficient condition for this implication to hold.

A similar question that I might post separated is related to the fact that symmetric lax monoidal functors not only preserve monoids on the category but also operads on the category (here I am using the monoidal product of the underlying category, not the composite, which is eventually defined in terms of the monoidal product of the underlying category). So a natural question again is whether a functor that preserves operads and their morphisms is automatically symmetric lax monoidal.

Question.Is there a lax monoidal functor $(F,\eta,\mu) : (\mathcal{C},\otimes) \to (\mathcal{D},\otimes)$ (with underlying functor $F$) such that $\mathrm{Mon}(F,\eta,\mu) : \mathrm{Mon}(\mathcal{C},\otimes) \to \mathrm{Mon}(\mathcal{D},\otimes)$ is equal to $F'$? $\endgroup$2more comments