Free objects and Gröbner-Shirshov bases in operated contexts

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