Let A be an abelian category and D the category having two objects and only one nonidentity morphism between them.
The functor category A$^D$ is also abelian and it is called an arrow category with objects morphisms in A and morphisms commutative squares.
I cannot see the equivalence between the functor category and the arrow category. I understand arrow category but how it is equivalent to the functor category? Any help would be appreciated!
Best Answer
Let $0$ and $1$ denote the objects of $D$ and write $a:0\rightarrow 1$ for the only non-identity arrow of $D$. To every functor $F:D\to A$ associate the morphism $F(a):F(0)\to F(1)$ in $A$. Conversely, to every morphism $f:X\to Y$ in $A$ associate the functor $\hat f:D\to A$ given by $\hat f(0)=X$, $\hat f(1)=Y$ and $\hat f(a)=f$. Can you continue from here?