## On the general coherence result of Power

### Steve Lack – 10 September 1997

In John Power's paper A general coherence result, JPAA 57(1989):165-173 it is proved that for a 2-monad T on Cat which preserve bijections on objects, every pseudo-T-algebra is equivalent to a strict T-algebra in the 2-category Ps-T-Alg of pseudo-T-algebras, (pseudo) T-morphisms, and T-transformations.
Here we strengthen this result, showing that the inclusion in Ps-T-Alg of its sub-2-category T-Alg_s comprising only the strict T-algebras and the strict T-morphisms has a left (2-)adjoint, and that the unit of this adjunction is an equivalence.
In fact for any 2-monad T on a cocomplete 2-category K, this adjunction exists if T is accessible (or has a rank), and we propose that ``the cohenrence theorem for T-algebras'' should mean that the unit of this adjunction is an equivalence. It is still not clear in what generality this theorem might be expected to hold.

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