According to <https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid> we can think of delooping of a group as the one object groupoid $BG$ consisting of a single object and whose morphisms are the elements of the group $G$ and the composition of morphisms are given by the group multiplication.


Now let us consider a category $C$ with a group object $\bar{G}$.


Now if $C$ has a forgetful functor to $Sets$ then there is a natural way to associate a one object groupoid to the group object (For example the way we talk about delooping of a Topological group, Lie group..etc).


**My question is the following:**


Let $D$ be a category with a group object $G'$(**without any other further assumptions**). Is there an analogous way to associate a one object groupoid to $G'$ ?


Or is there any necessary **and /or** sufficient condition for doing so?


**Apology in advance if my question is not making much sense.**



Delooping of a group object as a one object groupoid
To provide a constructive answer, suppose $C$ is a category with finite products, and $G$ is a group object in $C$. Then the one-object groupoid object $G\rightrightarrows \ast$ in $C$ really is the delooping of $G$ in the 2-category $\mathbf{Gpd}(C)$ of groupoid objects1 in $C$. That is, there is a 2-commuting square making $G$ the "[loop space object](https://ncatlab.org/nlab/show/loop+space+object)" of $G\rightrightarrows \ast$ at the canonical basepoint.




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1 A groupoid object in a category without all finite limits is the same as in a finitely-complete category, just that one must demand that the relevant pullbacks exist, or alternatively, one supplies explicit objects of composable pairs and triples. If this makes you uncomfortable, assume $C$ has all finite limits instead. Personally, I would go so far as to only require a terminal object exists, in order to define group objects.


