A universal homological $\delta$ functor $S_i$ has a universal property that for any universal $\delta$ functor $T$, and a natural transformation $T_0 \to S_0$, there is a unique extension.
A cohomological similiarly flips the arrows.
Can someone give me a counter-example showing we can't expect a cohomological $\delta$ functor to have the universal property of a homological one? I.e why in the cohomological we really only hope to extend $S_0 \to T_0$ and not the other way around.