The forgetful functor FiltSet to Set does not have a left adjoint, and egregiously so—you have added just enough structure to rule out free filtered sets, and may want to make note of where this is important..
The forgetful functor FiltSet to Set does not have a left adjoint, and egregiously so—you have added just enough structure to rule out free filtered sets, and may want to make note of where this is important..