The paper gives an unified formalism that encompasses the two most common mesh adaptation strategies: Hessian-based and goal-oriented. The first one is based on the control of the interpolation error of a solution field. The second one relies on the control of the approximation error of a scalar-output functional. Both of them have been widely used in aeronautics and derived in an anisotropic context by using a metric-based approach. If Hessian-based mesh adaptation is completely generic, it does not account for discretization error of the PDE at hand, contrary to the goal-oriented approach. The scope of this paper is to extend metric-based mesh adaptation to control a norm of the approximation error. This allows us to control simultaneously multiple functionals of interest as lift, drag, moment, without the need to solve multiple adjoint states. The procedure is based on the derivation of a corrector term that is then used as a functional for adjoint based-mesh adaptation. The estimate is derived within the continuous mesh framework, yielding naturally a fully anisotropic estimate.