Stable natural cohomology operation

Let $A$ and $B$ be abelian groups, and fix integers $n$ and $r$. A cohomology operation of type $n$ and degree $r$ is a natural transformation

$${\mathrm{\Phi }}_n\text{\hspace{0.17em}⁣}:{\tilde{H}}^n(-;A)⟶{\tilde{H}}^{n+r}(-;B)$$

of contravariant functors on based CW complexes. Thus every based map $f\text{\hspace{0.17em}⁣}:X\to Y$ gives the commutative naturality square

$$f^{\ast }{\mathrm{\Phi }}_n(y)={\mathrm{\Phi }}_n(f^{\ast }y)\mathrm{.}$$

A stable natural cohomology operation of degree $r$ is a sequence $\mathrm{\Phi }=({\mathrm{\Phi }}_n{)}_{n\in \mathbb{Z}}$ of such operations for which cohomology suspension commutes with $\mathrm{\Phi }$: for every based CW complex $X$ and every $x\in {\tilde{H}}^n(X;A)$,

$$\sigma ({\mathrm{\Phi }}_n(x))={\mathrm{\Phi }}_{n+1}\left(\sigma x\right)\in {\tilde{H}}^{n+r+1}(\mathrm{\Sigma }X;B)\mathrm{.}$$

Additivity is not part of this definition. Reduced cohomology is used so that the suspension condition has one uniform form, including degree zero. For the one-point based space every reduced group displayed above is zero, so both naturality and stability hold uniquely; no choice principle is used.