I want to learn spectral sequences and I'm helping myself with Vakil's book (http://math.stanford.edu/~vakil/216blog/FOAGdec3014public.pdf section 1.7).
Also, I probably should say that and I'm not used with Homological Algebra (which means in particular that I didn't do many exercises in the book, yet).
Anyway, at the end of page 60 Vakil proposes the reader the following "very worthwhile exercise": assume we have a double complex, that is a family of objects $\{E_0^{p,q}\}_{p,q\in\mathbb{Z}}$ together with morphisms $$d_\rightarrow:E_0^{p,q}\rightarrow E_0^{p+1,q}\;\;\text{ and }\;\;d_{\uparrow}:E_0^{p,q}\rightarrow E_0^{p,q+1}\;\;\;\;\;\forall p,q\in\mathbb Z$$ such that $d_\uparrow\circ d_\uparrow=d_\rightarrow \circ d_\rightarrow=0$ and (say) $d_\rightarrow\circ d_\uparrow=d_\uparrow d_\rightarrow$. Let $E_1^{p,q}$ be the homology of $E_0^{p,q}$ for $d_\rightarrow$. Then it is easy to see that $d_\uparrow$ induces a map $$d_1:=``d_\uparrow":E_1^{p,q}\longrightarrow E_1^{p,q+1},$$ which furthermore satisfies $d_1\circ d_1=0$. Then Vakil asks how one should define the map $$d_2: E_{2}^{p,q}\longrightarrow E_{2}^{p-1,q+2},$$ where $E_{2}^{p,q}$ is the homology of $E_1^{p,q}$ for $d_1$.
The construction should be similar to the one of the "linking" morphism of the long exact sequence, but here nothing's exact so I'm not sure how to "find preimages".