Those who have trouble accepting the objective reality of moral values and duties point to the fact that we can't be sure that these values are actually objective. They might just as easily be subjective from person to person. But for every argument that can be given to dispute the existence of objective moral values, a parallel argument against the external world can be given. Moral values seem to force themselves upon us. Our moral sense apprehends these moral realities, such as it is always wrong to torture a little baby and always right to protect that baby, whether we want it to or not. Likewise, we apprehend an external world through our senses of touch, smell, sight, etc whether we want to or not. It imposes itself on us. The person who wants to ask for justification for believing in objective morals should consider this parallel with the external world. Just as there is no test or experiment to get outside our physical senses and prove 100% that we aren't actually a brain in a vat or in the Matrix, likewise there is no test that we can perform to get outside our moral sense and see if the moral values and duties it apprehends are real. Moral values are self-evident, just as the external world is. We apprehend them through our senses and there is no reason to question the reliability of these senses. The belief in objective moral values is properly basic, just as the belief in the external world is properly basic.
I argue the presuppositions that one has to hold to deny the objective reality of the external world or of objective moral values and duties require more justification than do those of the one that accepts these realities. In order to question the reality of the external world, one has to assume that our senses are not reliable and the input we are receiving is potentially false. Likewise with objective morals, one has to assume that their moral sense isn't reliable. In the absence of defeaters for the acceptance of their reliability, there is no reason to think our physical or moral senses are faulty.