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.
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