TITLE: What's the (monoidal) image of a monoidal functor?
QUESTION [8 upvotes]: For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that image.
Is there anything vaguely like the image of a strong monoidal functor, which should be monoidal again?
What certainly doesn't work straightforwardly is just taking the image of the underlying functor and somehow putting a monoidal structure on it. Given $X, Y\colon \mathcal{C}$, how would you define $FX \otimes FY$? If you take the tensor product in $\mathcal{D}$, the result might not be in the image of $F$, but neither can you define it to be $F(X \otimes Y)$, since $F$ might no be injective on objects (so you can't find out a unique object to start with, and there is no canonical way to choose).
I'm utterly surprised I've never encountered such a construction. The only thing I've come across is for the case of fusion categories. There, the full subcategory spanned by summands of objects in the image of $F$ can be defined.
The only thing I can come up with for plain monoidal categories is the category spanned by the image of the underlying functor and all isomorphic objects. But this doesn't have good properties if I want to extend structures on $\mathcal{C}$ to $\mathcal{D}$. For example, if I have a braiding $c_{X,Y}\colon X \otimes Y \to Y \otimes X$ on $\mathcal{C}$, I can transport it onto objects of the form $FX$, but not onto all objects isomorphic to an $FX$, since I don't know which isomorphism to transport it along.

REPLY [4 votes]: Let $F : \mathcal{C} \to \mathcal{D}$ be a strong monoidal functor. Since monoidal categories are non-strict structures in the sense that e.g. associativity of $\otimes$ only holds up to isomorphism, it makes sense to look at those objects of $\mathcal{D}$ which are isomorphic to objects of the form $F(X)$, where $X$ is an object of $\mathcal{C}$. These constitute a full subcategory of $\mathcal{D}$, which contains the unit object and is closed under tensor products, hence inherits a monoidal structure.
If (only) $\mathcal{C}$ has a braiding, there is no reason to expect that it induces a braiding on the image, since this image depends on $F$. For this, we should require that $\mathcal{D}$ is also braided and that $F$ is a braided monoidal functor. It follows that the image, as defined above, carries a braiding induced by $\mathcal{D}$, and that $F$ factors as a braided monoidal functor into it.