For any integer $n$, let $h^n:\sf Top^{op}_*\to Ab$ be functors satisfying the axioms of a reduced generalized cohomology theory, as defined in Yiannis Loizides' paper on Atiyah-Hirzebruch spectral sequence (https://math.gmu.edu/~yloizide/Atiyah-Hirzebruch.pdf).
If $X$ is a CW-complex and $X_p$ its $p$-skeleton for any integer $p$, the inclusion $X_{p-1}\to X_p$ yields a long exact sequence $$\dots \to h^{n-1}(X_{p-1})\xrightarrow{\beta}h^n(X_p/X_{p-1})\xrightarrow{\gamma} h^n(X_p)\xrightarrow{\alpha} h^n(X_{p-1})\to \dots$$ which after changing the index $n$ with $p+q$, for integer $q$, becomes as below. $$\dots \to h^{p+q-1}(X_{p-1})\xrightarrow{\beta}h^{p+q}(X_p/X_{p-1})\xrightarrow{\gamma} h^{p+q}(X_p)\xrightarrow{\alpha} h^{p+q}(X_{p-1})\to \dots$$ The paper then defines an exact couple like this:
So I would say that the map $D\to D$ in the triangle is defined as $$h^{p+q}(X_p)\xrightarrow{\alpha} h^{p+q}(X_{p-1})\hookrightarrow D$$ on each component $D^{p,q}$. However I don't understand the meaning of $(1,1)$ above $\alpha$ in the triangle; if we had two maps $\alpha,\alpha':D\to D$ I would interpret this as $\alpha-\alpha'$, but the only map $D\to D$ that I see is the one I just talked about.
The $(-1,1)$ indicates the degree, i.e. $\alpha$ maps the summand $D^{p,q} = h^{p+q}(X_p)$ into $h^{p+q}(X_{p-1})=D^{p-1,q+1}.$ Analogously for the other maps.