Induced Monoidal Structure by Isomorphism

Let $$\mathscr{C}$$ and $$\mathscr{D}$$ be a pair of categories and assume they are isomorphic via a functor $$F \colon \mathscr{C} \longrightarrow \mathscr{D}.$$ Suppose that $$(\mathscr{D}, \otimes, 1, \sigma)$$ has a symmetric monoidal structure (not necessarily strict), where $$1$$ is the unit and $$\sigma$$ is the braiding. Then, it seems natural to think that such structure can be transported to $$\mathscr{C}$$, defining $$x\Box y := F^{-1}(F(x)\otimes F(y))$$ for every $$x,y \in \operatorname{obj}(\mathscr{C})$$, $$f\Box g:= F^{-1}(F(f)\otimes F(g))$$ for every $$f,g \in \operatorname{Hom}(\mathscr{C})$$ and similarly for the rest of the monoidal structure. However, I'm not sure about if this really yields a monoidal structure on $$\mathscr{C}$$. Is this result (or similar) proven in some reference or something? Thanks in advance!

• Yes, this is easy to prove if $F$ is an isomorphism ($F$ just preserves every monoidal category axiom precisely; this is "transport of structure"). An isomorphism will even send strict monoidal structures to strict monoidal structures. The interesting statement is that this remains true if you only require that $F$ be an equivalence; then $F$ will no longer preserve strict monoidal structures and this is what you need non-strict monoidal structures for. Commented Sep 14, 2020 at 19:12
• Is there any proof of the equivalence case in the mathematical literature? Commented Sep 14, 2020 at 20:32
• Probably? Here's an MO thread about it: mathoverflow.net/questions/168351/… Commented Sep 14, 2020 at 20:50
• Does this answer your question? math.stackexchange.com/questions/583042/… Commented Sep 15, 2020 at 0:58