I am studying homological algebra and I am completely lost in sign conventions, I need someone to checjk the definitions I am using. Unfortunately the lecturer of my course is not very precise on this matter and we don't follow the notations of any book. In the course I have followed the following definitions where given for a complex $A$ in an abelian category. Shift functor: $$A[p]_n=A_{n-p}$$
It was not defined on arrows, I suspect that the lecturer wants a $(-1)^p$ sign in front of the differentals but I am not sure.
Cone and cocones: let us stick for a moment on cones and cocones of complexes, that is, $\mathrm{cone}(A)=\mathrm{cone}(\mathrm{id}_A)$. The cone has been defined as $$ \mathrm{cone}(A)_n=A_n \oplus A_{n-1}, \quad d_n=\begin{pmatrix} d_n & (-1)^n \\ 0 & d_{n-1} \end{pmatrix}$$ and the cocone $$ \mathrm{cocone}(A)_n=A_n \oplus A_{n+1}, \quad d_n=\begin{pmatrix}(-1)^n d_n & 0 \\ 1 & (-1)^{n+1}d_{n+1} \end{pmatrix}.$$
Now it is claimed that $\mathrm{cocone}(A)=\mathrm{cone}(A)[-1]$. But this does not seem to work since with the definitions above I would have $$\mathrm{cone}(A)[-1]=C_{n+1}\oplus C_{n}, \quad d_n=\begin{pmatrix} -d_{n+1} &(-1)^{n}\\ 0 & -d_n\end{pmatrix}$$ and this does not coincide with the cocone, I have tried defining some isomorphism $C_{n+1} \oplus C_{n} \to C_{n} \oplus C_{n+1}$ with $$\begin{pmatrix} 0 &(-1)^{\eta(n)} \\ (-1)^{\epsilon(n)}& 0 \end{pmatrix}$$
where $\eta$ and $\epsilon$ wuold take care of the signs, but nothing seems to work. what am I missing?
