I was given the following exercise, but the person who gave me the exercise wasn't sure about some of the details (such as signs).
Let $(C, \partial)$ be a chain complex, and $h: (C, \partial) \rightarrow (C, \partial)$ satisfy $h^2 = 0$. Then $(C, \partial) \cong (C, \partial \pm h\partial \pm \partial h)$.
I am not sure if $\cong$ is supposed to be an isomorphism or a chain homotopy. The symbol suggest isomorphism, but I am not sure if we are working in the category of chain complexes up to homotopy.
Using $h^2 = 0$ gives us that $(C, \partial \pm h\partial \pm \partial h)$ is indeed a chain complex. If the signs on $h\partial$ and $\partial h$ differ, then the isomorphism is trivial as $\partial h = h \partial$. Thus I believe that the signs are both $+$. If somebody knows precisely the statement of the problem, that would help me do the exercise.