The definition of "union" in Abelian Category The following  is a passage, which introduces the definition of regular spectral sequence,  from a book.
Suppose in the Abelian category $\mathcal{C},$ the direct sum of any family of objects exists. Then for any object $A$ in $\mathcal{C}$ and any family $A_{i}(i \in I)$ of sub-objects of $A$
we define $\bigcup_{i \in I} A_{i}$ to be the image of the canonical morphism $\bigoplus_{i \in I} A_{i} \rightarrow A .$
My question : 1. In a general Abelian category,  the direct sum of any family of objects may not exists, is it? Can one give some examples?


*Why do we define the  $\bigcup_{i \in I} A_{i}$ as above? Why cannot define it set theoretically?

Any answers is welcome!
 A: I don’t know about spectral sequences yet, but I hope the following clears things up a bit.

*

*@AnginaSeng gave an example ($\mathsf{FinVect}_K$) of an abelian category lacking arbitrary coproducts. Maybe you want to restrict to finite families of subobjects or require the abelian category to be cocomplete?


*Not all abelian categories have objects with underlying sets, e.g. the category $\mathsf{Ch}(\mathcal{A})$ of chain complexes on an abelian category $\mathcal{A}$. Hence defining things set-theoretically is not really possible. However, drawing intuition from the exact sequence
$$0 \rightarrow \bigcap_i V_i \rightarrow \bigoplus_i V_i \rightarrow \langle \bigcup_i V_i \rangle \rightarrow 0$$
in vector spaces, modules and alike it makes sense to define the subobject spanned by the family of subobjects as the image of the canonical map $\bigoplus_i V_i \rightarrow V$.
Note I like to reserve the notation $\bigcup$ for sets, as I find writing $\bigcup V_i$ confusing in the case of vector spaces etc. This is why I added a span $\langle \cdot \rangle$ symbol in my answer.
