  This paper investigates algebraic objects equipped with an operator, such as
operated monoids, operated algebras etc. Various free object functors in these
operated contexts are explicitly constructed. For operated algebras whose
operator satisfies a set $\Phi$ of relations (usually called operated
polynomial identities (aka. OPIs)), Guo defined free objects, called free
$\Phi$-algebras, via universal algebra. Free $\Phi$-algebras over algebras are
studied in details. A mild sufficient condition is found such that $\Phi$
together with a Gr\"obner-Shirshov basis of an algebra $A$ form a
Gr\"obner-Shirshov basis of the free $\Phi$-algebra over algebra $A$ in the
sense of Guo et al.. Ample examples for which this condition holds are
provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs,
averaging OPIs and Reynolds OPI.
