Codescent objects and coherence

We describe 2-categorical colimit notions called codescent objects of coherence data, and lax codescent objects of lax coherence data, and use them to study the inclusion, T-Alg_s → Ps-T-Alg, of the 2-category of strict T-algebras and strict T-morphisms of a 2-monad T into the 2-category of pseudo T-algebras and pseudo T-morphisms; and similarly the inclusion T-Alg_s → Lax-T-Alg_ℓ, where Lax-T-Alg_ℓ has lax algebras and lax morphisms rather than pseudo ones. We give sufficient conditions under which these inclusions have left adjoints. We give sufficient conditions under which the first inclusion has left adjoint for which the components of the unit are equivalences, so that every pseudo algebra is equivalent to a strict one.