Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend $$\int^{i \in I} D(i,i)$$ can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ modulo the equivalence relation generated by $$D(i,i) \ni D(f,\mathrm{id}_i)(x) \sim D(\mathrm{id}_j,f)(x) \in D(j,j)$$ for $f : i \to j$, $x \in D(j,i)$.
Question. Under which conditions on $I$ and $D$ is every element of $\int^{i \in I} D(i,i)$ the class of some element in some $D(i,i)$? In other words, when $U : \mathsf{Ab} \to \mathsf{Set}$ is the forgetful functor, I ask for a condition that the canonical map $$\int^{i \in I} U(D(i,i)) \to U\left(\int^{i \in I} D(i,i)\right)$$ is surjective. This is clearly equivalent to the condition that the image of this map is closed under sums.
For example, this is the case when for all $x \in D(i,i)$, $y \in D(j,j)$ there is some span $i \leftarrow k \rightarrow j$ such that $x$ has a preimage in $D(i,k)$ and $y$ has a preimage in $D(j,k)$, since then $x$ and $ y$ are equivalent to elements in $D(k,k)$. This condition is already useful in practice, but obviously it is quite strong. Probably weaker assumptions are also sufficient.
Answers for the following special case are appreaciated as well: Let $A : I^{\mathrm{op}} \to \mathsf{Ab}$, $B : I \to \mathsf{Ab}$ be two functors and let $D(i,j) = A(i) \otimes B(j)$, so that the coend is the tensor product of functors $A \otimes_I B$. In this case, the question is when every element of $A \otimes_I B$ is the class of some element in some $A(i) \otimes B(i)$.
Background. I have to do some computations with some complicated coends, and it won't be sufficient to use the universal property. I really need a better idea of the element structure of the coend. The next step would be to find a description of the kernel of $D(i,i) \to \int^{i \in I} D(i,i)$.
