Understanding Functor Categories in Graded Abelian Groups

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Homework Statement

Let \mathbb N be the discrete category of natural numbers. Describe the functor category \mathrm{Ab}^{\mathbb N} (commonly known as the category of graded abelian groups).

Homework Equations

The Attempt at a Solution

Since \mathbb N is discrete, a functor \mathbb N\xrightarrow A\mathrm{Ab} is simply a sequence (A_n) = A_0,A_1,\dots of abelian groups; an arrow (A_n)\xrightarrow{\sigma} (B_n) is given by arrows A_0\xrightarrow{\sigma_0}B_0, A_1\xrightarrow{\sigma_1}B_1,\dots.

This looks right, but seems too simple to me. I don't know very much about grading, but I thought there had to be some way of "going up the A's" (like \otimes\colon V^{\otimes i}\times V^{\otimes j}\to V^{\otimes i+j} in the case of the tensor algebra)

 

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For a graded ring you need to have an i thing times a j thing be an i+j thing. Since this problem is just with abelian groups, there is no such condition.