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 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.

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.