# The Unapologetic Mathematician

## 2-functors

Of course along with 2-categories, we must have 2-functors to map from one to another.

So, what’s a 2-functor? Since we defined a 2-category as a category enriched over $\mathbf{Cat}$, a 2-functor should be a functor enriched over $\mathbf{Cat}$. That is, it consists of a function on objects and a functor for each hom-category, each of which consists of a function on 1-morphisms (the objects of the hom-category) and a function for each set of 2-morphisms. Then there are a bunch of relations.

Let’s expand this a bit. A 2-category $\mathcal{C}$ has a collection $\mathrm{Ob}(\mathcal{C})$ of objects, a collection $\mathrm{Ob}(\hom_\mathcal{C}(A,B))$ of 1-morphisms for each pair $(A,B)$ of objects, and a collection $\hom_{\hom_\mathcal{C}(A,B)}(f,g)$ of 2-morphisms for each pair $(f,g)$ of 1-morphisms between the same pair of objects. And all the same remarks go for another 2-category $\mathcal{D}$.

So a 2-functor $F$ has

• a function $F:\mathrm{Ob}(\mathcal{C})\rightarrow\mathrm{Ob}(\mathcal{D})$
• for each pair $(A,B)$ of objects of $\mathcal{C}$, a functor $F:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{D}(F(A),F(B))$, each consisting of
• a function $F:\mathrm{Ob}(\hom_\mathcal{C}(A,B))\rightarrow\mathrm{Ob}(\hom_\mathcal{D}(F(A),F(B)))$
• for each pair $(f,g)$ of 1-morphisms from $A$ to $B$ a function $F:\hom_{\hom_\mathcal{C}(A,B)}(f,g)\rightarrow\hom_{\hom_\mathcal{D}(F(A),F(B))}(F(f),F(g))$

Now the composition functors $\circ$ give us functions for composing 1-morphisms and “horizontally” composing 2-morphisms, and the hom-categories give us functions $\cdot$ for “vertically” composing 2-morphisms. For each object we have an identity 1-morphism, and for each 1-morphism we have an identity 2-morphism. A 2-functor will preserve all these structures. First of all, since there are functors between the hom-categories the vertical composition is preserved, along with the identity 2-morphisms. The diagrams for enriched functors say that the identity 1-morphisms, the composition of 1-morphisms, and the horizontal composition of 2-morphisms are all preserved.