The Stacks project

Remark 12.16.3 (Warning). There are abelian categories $\mathcal{A}$ having countable direct sums but where countable direct sums are not exact. An example is the opposite of the category of abelian sheaves on $\mathbf{R}$. Namely, the category of abelian sheaves on $\mathbf{R}$ has countable products, but countable products are not exact. For such a category the functor $\text{Gr}(\mathcal{A}) \to \mathcal{A}$, $(A^ i) \mapsto \bigoplus A^ i$ described above is not exact. It is still true that $\text{Gr}(\mathcal{A})$ is equivalent to the category of graded objects $(A, k)$ of $\mathcal{A}$, but the kernel in the category of graded objects of a map $\varphi : (A, k) \to (B, k)$ is not equal to $\mathop{\mathrm{Ker}}(\varphi )$ endowed with a direct sum decomposition, but rather it is the direct sum of the kernels of the maps $k^ iA \to k^ iB$.