Whats the generalisation in category theory of the classical cover in topology?

In topology a cover of a space is a set of subspaces whose union is the space. Obviously a subspace is an inclusion.

A space is an object in the category $Top$.

In category theory, according to nlab, a cover of an object $U$ (of some category) is a family of morphisms $f_i:U_i\rightarrow U$. It doesn't look as though they require that the 'union' of the $U_i$ needs to be $U$. Nor that they should be 'inclusions'. But if we do require it, what is the best way to generalise the condition? Is there standard nomenclature for this?

The inclusion of a subspace $U_i$, is probably best generalised by requiring that each $f_i$ is monic.

For the 'union' of the $U_i$, can we simply say that the maps are jointly epic? That is whenever $g\circ f_i=h\circ f_i$ for all $i$, we have $g=h$. But this simply says that $U$ is a coproduct of the family $f_i$.

Or alternatively, in topology, we can take the disjoint union of the cover and this will be surjective and locally injective - that is a local homeomorphism. And this is also called an etale mapping.

If the category has all pushouts and an initial object $0$, we can construct the disjoint pushout $\sqcup g_i$ by the initial morphisms $g_i:0\rightarrow U_i$, and then by the pushout property we have $\sqcup g_i:\sqcup U_i\rightarrow U$ that is epic. But in the presence of initial objects pushouts & coproducts can construct each other. So the alternative is actually no real alternative.

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Not every jointly epic family is a coproduct diagram ... –  Martin Brandenburg Jan 5 at 23:41
@Brandenburg: ok, its the reverse that is true. Is there a nice condition that makes them equivalent? Actually - I don't think there is, its the 'existence' in the coproduct thats the the problem not the 'uniqueness'. –  Mozibur Ullah Jan 6 at 0:00
In $\mathsf{Set}$, $\{X_i \to X\}$ is a coproduct diagramm if it is jointly epic, each $X_i \to X$ is monic, and $X_i \times_X X_j = 0$ (initial) for all $i \neq j$. Probably the same holds for large classes of categories (regular categories?). –  Martin Brandenburg Jan 6 at 0:28
@MartinBrandenburg You should probably assume extensiveness as well. Then it becomes a question that can be solved by descent theory. –  Zhen Lin Jan 6 at 2:03