Creation of limits up to isomorphism

An equivalence-invariant lifting property permits the underlying cone of the lift to be identified with the chosen cone by an isomorphism, rather than requiring literal equality. Lifts are unique up to a unique isomorphism compatible with that identification. A monadic adjunction defined by equivalence of its comparison functor has this property by monad algebra forgetful functor creates limits. Literal creation follows if the comparison is an isomorphism over the base, or suitable strict transport of objects is part of the convention. An equivalence alone does not guarantee a literally prescribed underlying object lies in the image.