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.**

Everyobject in anarbitrarycategory gives a one-object groupoid: its automorphism group. Since a one object groupoid is exactly the same info as a group, you might as well phrase your question as: given a category C, a group object G in C, how do I associate to G (presumably functorially) a plain group G' (i.e. a group object in Set)? $\endgroup$ – David Roberts Oct 18 at 7:43internal toC. You must distinguish between a groupoid object in C, and a groupoid. It's not clear to me which one one you want, now, looking at the original question and your comments. "Is the groupoid object ... you mentioned in the previous comment....delooping ?" Yes. $\endgroup$ – David Roberts Oct 18 at 9:05thinkit's going to be a delooping in the nonabelian derived category of your ambient category, but I haven't thought this through). @DavidRoberts I just didn't want to spend time figuring out what a groupoid object is if you don't have enough pullbacks (I think the correct strategy is to work in $\mathcal{P}_\Sigma$, but I haven't thought it through) $\endgroup$ – Denis Nardin Oct 18 at 9:12