Operad algebra

In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring "R", with an operad replacing "R".
Definitions.
Given an operad "O" (say, a symmetric sequence in a symmetric monoidal ∞-category "C"), an algebra over an operad, or "O"-algebra for short, is, roughly, a left module over "O" with multiplications parametrized by "O".
If "O" is a topological operad, then one can say an algebra over an operad is an "O"-monoid object in "C". If "C" is symmetric monoidal, this recovers the usual definition.
Let "C" be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If formula_1 is a map of operads and, moreover, if "f" is a homotopy equivalence, then the ∞-category of algebras over "O" in "C" is equivalent to the ∞-category of algebras over "O"' in "C".