When are cohomology operations determined by their action on coefficients?

It is well-known that K-theory operations are determined by the action on coefficients, but I don't know the right way to prove this fact, nor a reference for the same. On the other hand, clearly this is not true for ordinary cohomology operations. In general, when will this be true, and how does one prove it?

Does this make sense to ask also even for unstable operations?

• Are you asking when the map $\pi_*F(E,E)→\mathrm{Hom}_{\pi_*\mathbb{S}}(\pi_*E,\pi_*E)$ is injective? You can interpret it as a statement about the universal coefficient spectral sequence (i.e. that everything above the 0-line dies) but I doubt it's all that helpful... – Denis Nardin Feb 26 at 16:24
• @DenisNardin maybe the OP means something slightly different since most interesting operations on K-theory are not linear, hence not represented by maps $K\to K$. – Dylan Wilson Feb 26 at 16:26
• @DylanWilson Right, but in that case why are they talking about unstable operations as if they were a different thing? – Denis Nardin Feb 26 at 16:28
• i guess the main question is about stable, hence linear, operations, and then a secondary question is whether we can also say anything about the non-additive ones. – xir Feb 26 at 16:31
• @DylanWilson This is indeed true for MU, see Lemma 4.1.15 of the green book. – skd Feb 26 at 18:15