Why does the 2-category of groups have (some, strict) coinserters but not (strict) inserters?

Question

Let me begin by mumbling some abstract nonsense, and then attempt to be concrete. The category of groups inherits the structure of a strict 2-category from the 2-category of small categories. Explicitly, a 2-morphism between $\varphi$ and $\varphi'\colon H \to G$ is an element $g \in G$ such that for each $h \in H$, we have $\varphi(h) = g\varphi(h)g^{-1}$. I'm curious why this category has coinserters but not inserters.

So much for the mumbling. Let $\varphi$ and $\varphi'\colon H \to G$ be homomorphisms of groups. Let $\psi\colon K \to H$ be a homomorphism and $g$ an element of $G$. We say that the pair $(\psi,g)$ inserts $\varphi$ and $\varphi'$ if we have that for all $k \in K$, the following equality holds in $G$

$$\varphi\psi(k) = g\varphi'\psi(k)g^{-1}.$$

The collection of pairs $(\psi,g)$ with domain $K$ that inserts $\varphi$ and $\varphi'$ forms a category: a morphism in this category from $(\psi,g)$ to $(\psi',g')$ is an element $h \in H$ such that for all $k$ in $K$, we have the equality

$$\psi(k) = h\psi'(k)h^{-1}$$

as well as the equality

$$\varphi(h)g' = g\varphi'(h).$$

A strict inserter of $\varphi$ and $\varphi'$ would be a group $\mathrm{Ins}(\varphi,\varphi')$ equipped with an inserting pair $(\Psi,g)$ which is universal for this property in the following sense: If $(\psi',g')$ is an inserting pair with domain $K$, there exists a unique homomorphism $\eta\colon K \to \mathrm{Ins}(\varphi,\varphi')$ such that $\psi' = \Psi\eta$ and such that $g = g'$. This latter requirement kills the possibility of strict inserters: even if $G$ is abelian, so that the condition of a pair $(\psi,g)$ inserting $\varphi$ and $\varphi'$ is just the condition that $\psi$ equalizes $\varphi$ and $\varphi'$, the free choice of $g$ makes it impossible to have a universal choice of $g$.

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The dual notion is of a coinserter. This is a group $K$ equipped with a homomorphism $\Psi\colon G \to K$ and an element $t \in K$ such that $\Psi\varphi(h) = t\Psi\varphi'(h)t^{-1}$ for all $h \in H$. If a pair $(\psi',t')\colon G \to K'$ coinserts $\varphi$ and $\varphi'$, there must be a unique homomorphism $K \to K'$ satisfying the obviously dual condition to the above. Now since $t$ and $t'$ live in different groups, suddenly we're able to construct coinserters, certainly at least in the case where $\varphi$ and $\varphi'$ are injective, but I imagine a variant of the construction produces a coinserter in all cases.

Explicitly the coinserter is the pair $(\iota, t)\colon G \to G*_H$, where $G*_H$ denotes the HNN extension of $G$ with associated subgroups $\varphi(H)$ and $\varphi'(H)$, $\iota$ is the canonical inclusion of $G$ into $G*_H$ and $t$ is the "stable letter" for the HNN extension.

A presentation for $G*_H$, given that a presentation for $G$ is $G = \langle S \mid R \rangle$ is as follows:

$$\langle S, t \mid R, t\varphi(h)t^{-1} = \varphi'(h) \rangle$$

as $h \in H$ varies. It should be clear that if a pair $(\psi,t')\colon G \to K$ coinserts $\varphi$ and $\varphi'$, we can define a homomorphism $G*_H \to K$ taking $\iota(g)$ to $\psi(g)$ for all $g \in G$ and taking $t$ to $t'$.

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Sorry for the ramble. Anyway: is there some deeper reason that the 2-category of groups should have (strict) coinserters but not (strict) inserters? After all, the 2-category of categories has both.

Answer 1

One viewpoint goes as follows: the 2-categorical structure on groups can be seen as coming from inner automorphisms, so that a 2-cell is given by an inner automorphism that translates one map to the other. Now, inner automorphisms of an object can be defined in any category (see e.g. this paper) using the notion of isotropy group, a particular functor from the starting category to groups.

Moreover, one can use these abstract inner automorphisms to promote any category into a 2-category (albeit not in a functorial way: a given functor might not become a 2-functor). Me and Pieter Hofstra have some results on how 2-categorical limits and colimits behave in the resulting 2-category. In particular, as soon as your starting category is finitely cocomplete, it has all coinserters iff the isotropy functor is representable. Moreover, all limits and connected colimits of the underlying category satisfy the required two-dimensional universal property as well. However, inserters, equifiers or 2-d coproducts exist iff the isotropy is trivial. For what it's worth, I've given a talk about this stuff here but there's no publicly available writeup yet.

One can debate if this counts as a "deep reason", as ultimately the general proof that strict inserters do not exist boils down to the kind of situation you considered. That said, this does put the observation in context, so perhaps it still counts.