Can the category of chain complexes in an additive category $A$ be realized as as subcategory of presheaves on $\mathbb{Z}$? Here I am thinking of $\mathbb{Z}$ as a poset category. I am asking a different question than this one since I have a specific domain in mind.
Here is why I think this should work: by definition, a chain complex is given by a sequence $ \{C_n\}_{n\in \mathbb{Z}}$ with morphisms $\{\partial_n : C_n \to C_{n-1}\}$ such that $\partial_n \circ \partial_{n+1}=0$.
We can construct a functor out of this data as follows: define $C:\mathbb{Z}^{\text{op}} \to A$ by setting $C(n):= C_n$. For the unique $f_n : n \to n-1$, set $C(f_n) := \partial_n$. We should be able to send the unique $f : n \to k$, for $k<n-1$, to $0$ by the exactness property of chain complexes.
In the other direction, consider a functor $C : \mathbb{Z}^{\text{op}} \to A$ satisfying the property that $C(f)= 0$ for $f : n \to k$ with $k<n-1$. We should be able to recover a chain complex by setting $\{C_n\}_{n\in \mathbb{Z}}=\{C(n)\}_{n\in \mathbb{Z}}$ and $\{\partial_n\}_{n\in \mathbb{Z}}=\{C(f_n)\}_{n\in \mathbb{Z}}$, where $f_n : n \to n-1$ is the unique morphism. Then the exactness property is met by hypothesis.
In this way, we should be able to identify the category of chain complexes with a subcategory of presheaves on $\mathbb{Z}$. However, I did not find such a description in the link I gave for the definition.
Additionally, the question I link uses a much fancier domain category, one with a lot more morphisms. I think this does not matter since the "extra morphisms" will be recovered the domain from the fact that $A$ is additive. As far as I can tell, the use of a fancier domain category is just to get a sort of "free chain complex", which is nice since we do not have to restrict to a subcategory of functors in that case.
But these two things together lead me to believe I may have made a mistake in my reasoning. Does this look like a correct way to define the category of chain complexes?