How to Prove Distinct parallel morphisms define distinct natural transformations.

Question

Prove that distinct parallel morphisms $f,g: c \to d$ define distinct natural transformations $f∗,g∗: C(−,c)\Rightarrow C(−,d)$ and $f∗,g∗: C(d,−)⇒C(c,−)$ by post- and pre-composition. I have no idea how to do this. Please give me a hint. No solutions.

Answer 1

Suppose that $f_*$ and $g_*$ coincide. We can consider $C(c,c) \rightarrow C(c,d)$ and get $g = g_*(id_c) = f_*(id_c) = f$, a contradiction as $f$ and $g$ were distinct. Analogously we get what you want for the other one.