We consider a scalar-tensor theory of gravity wherein the scalar field $\phi$
    includes the gravitational coupling $G$ and the speed of light $c$, both of
    which are allowed to be functions of the spacetime coordinates. The effect of
    the cosmological coupling $\Lambda$ is accommodated within a possible behaviour
    of $\phi$. The dynamics of $\phi$ is analyzed in the phase space where we
    describe gravity from a precisely specified form of the action. For reasonable
    assumptions on the potential $V\left(\phi\right)$ and the matter-energy
    content, we show that an attractor point can be reached regardless the type of
    Hubble parameter functional behaviour. The phase space analysis is performed
    both with the linear stability theory and via the more general Lyapunov’s
    method. When the system gets to the stable point, the dynamics of $\phi$ ceases
    and the constraint $\dot{G}/G=\sigma\left(\dot{c}/c\right)$ with $\sigma=3$
    must be satisfied for the rest of the cosmic evolution. The latter result
    realizes our main motivation: To provide a physical foundation for the
    phenomenological model admitting
    $\left(G/G_{0}\right)=\left(c/c_{0}\right)^{3}$ used recently to interpret
    cosmological and astrophysical data. The thus co-varying couplings $G$ and $c$
    impact the cosmic evolution after the dynamical system settles to equilibrium.
    The secondary goal of our work is to investigate how this impact occurs. This
    is done by constructing the generalized continuity equation in our
    scalar-tensor model and considering two possible regimes for the varying speed
    of light\textemdash decreasing $c$ and increasing $c$\textemdash while solving
    our modified Friedmann equations. It is shown that cosmic evolution after the
    equilibrium accommodates radiation- and matter-dominated eras that naturally
    evolve to an accelerated expansion typical of dark energy.

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