Transporting algebraic structure along adjoint equivalences I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of view, at least). Given an (edit: equivalence invariant) algebraic structure on a groupoid $G$, say given by some 2-monad $M$ on the 2-category of groupoids, and an adjoint equivalence $f\colon X \to G$, consider the 2-groupoid of lifts of the algebraic structure along $f$, namely lifts of $f$ through the forgetful functor from the 2-category of $M$-algebras. Are there results that say this 2-groupoid is contractible? It seems very likely and also likely to be proven somewhere.
Secondly, I want to apply this to coherent 2-groups, which I guess are algebras for a 2-monad on groupoids, but would like to make sure.
 A: Here is another answer which is perhaps more "in the spirit" of the question. 
Suppose that $U: \mathcal{A} \to \mathcal{C}$ is a functor of 2-categories. Let $x$ and $y$ be objects of $\mathcal{C}$, and $f: x \to y$ a fixed morphism. Suppose that we have a lift $Y \in \mathcal{A}$ of $y \in \mathcal{C}$, that is an object such that $U(Y) = y$. 
We can define the 2-category $Lift(f; Y)$ of lifts of $f$ relative to $Y$. It is the 2-category whose objects are arrows $F: X \to Y$ such that $U(F) = f$, whose morphisms are the obvious triangles (with 2-isomorphisms witnessing the commutativity of the triangle). The top edge of the triangle maps via $U$ to the identity morphism of $x$. The 2-morphisms are maps of the top edges of triangles such that the obvious diagram commutes and such that the 2-morphisms map via $U$ to the identity 2-morphism of the identity 1-morphism of $x$. 
I think this is the 2-category of lifts that you meant, or perhaps you meant to restrict just to the invertible part of it. Either way...
Now suppose that:


*

*The functor $U$ is conservative in the following sense: a 1-morphism of $\mathcal{A}$ is an equivalence in $\mathcal{A}$ if and only if its image under $U$ is an equivalence in $\mathcal{C}$; a 2-morphism of $\mathcal{A}$ is an isomorphism if and only if its image under $U$ is an isomorphism; and two 2-morphisms of $\mathcal{A}$ are equal if and only if they are equal after applying $U$. (This is really an analog of being conservative and faithful). 

*You can lift your structure along equivalences and invertible transformations. Meaning that if $f:x \to y$ is in $\mathcal{C}$ and $Y \in \mathcal{A}$ lifts $y$, then there exists some lift of $f$ to $\mathcal{A}$. Moreover if $\alpha: f \cong g$ is a 2-isomorphism in $\mathcal{C}$ and you can lift $f$ and $g$ to parallel arrows in $\mathcal{A}$, then there exists some compatible lift of $\alpha$ to $\mathcal{A}$. 

*The morphism $f$ is an equivalence.
Then the 2-category $Lift(f; Y)$ is contractible. 
I already gave an example where $\mathcal{A}$ is the category of algebras for a 2-monad where there can fail to be lifts. 
However in your situation for coherent 2-groups you know that you can transfer the structure along equivalences and 2-isomorphism. Moreover we know that the forgetful functor from coherent 2-groups to groupoids is conservative in the sense I describe above. Hence the 2-category $Lift(f; Y)$ of lifts is contractible in this case. 
A: This is true in your example of coherent 2-groups. There are many sources that show that you can transfer monoidal structures along adjoint equivalences. You just need to adapt those to also transfer the coherent inverses, which should not be too difficult conceptually but might be tedious. 
However the general statement is false, I believe. 
Consider the 2-monad whose algebras are strict group objects. In other words you have a strict monoidal structure and strict inverses. Note that every coherent 2-group is equivalent to one of these. If it were true that you could transfer this structure along an adjoint equivalence, then this would imply that any coherent 2-group is equivalent to a 2-group which is both strict and skeletal. You first replace by a strict 2-group and then you apply the transference result (with respect to strict group objects) with X skeletal. 
I suspect that it might be true if your 2-monad satisfies some kind of cofibrancy (aka flexible) condition. Have you looked in Blackwell-Kelly-Power?
