Given
  R : D → C
  d : D
  c : C
  f : R(d) → c

Define
  R̂ : d/D → C
  R̂(u : d → d' in D) = R(d')

  d̂ : d/D
  d̂ = (id : d → d in D)

  f̂ : R̂(d̂) → c
  f̂ = f

Then you are asking for the last d̂' in d/D with some
  f̂' : R̂(d̂') → c

This is H(c), if H : C → d/D is right adjoint to R̂.