# Arrow between endofunctors over a symmetric monoidal category.

Consider an arrow between the categories of endofunctors over two symmetric monoidal (SM) categories $\mathcal{C}$ and $\mathcal{D}$

$$a:End(\mathcal{C}) \rightarrow End(\mathcal{D})$$

It is a functor acting over endofunctors over a SM category but also acts over natural transformations between endofunctors over a SM category.

Is $a$ a 2-functor? Which (higher order) category $a$ belongs to (seen as an arrow)? Does $a$ send objects of $\mathcal{C}$ to objects of $\mathcal{D}$ in some sense?

• There seems to be some confusion here. An arrow between two categories is usually a functor. A natural transformation is an arrow between functors. – Zhen Lin Apr 17 '14 at 9:28
• Edited, thanks. – gibarian Apr 17 '14 at 9:44

Any object $C \in \mathcal{C}$ can be seen as a constant functor sending everything to the terminal category containing $C$, so yes, the objects of $\mathcal{C}$ are being sent to the objects of $\mathcal{D}$. And any morphism in $\mathcal{C}$ is a natural transformation of constant functors (easy exercise). So your $a$ gives rise to a functor $a: \mathcal{C} \to \mathcal{D}$.
But there is more to it. Consider the bicategory of symmetric monoidal categories. It has two 1-object subcategories, with objects $\mathcal{C}$ and $\mathcal{D}$, respectively. Your $a$ is a bifunctor between them. This is what you get if you "deloop" your picture of the categories $\operatorname{End}\mathcal{C}$ and $\operatorname{End}\mathcal{D}$, they're actually 1-object bicategories.