I don't remember if I've already seen this question even here or in MO or in my mind. This is partly related to questions arose about differences between homology and cohomology; I'm wondering if some kind of difference can appear before computing $H^n(-)$, $H_n(-)$, looking at the bare complex of (co)chains...
My question: Given a chain complex $\mathcal C=\{C_n,\partial_n\}$, consider the image complex $\hom(\mathcal C,X)=\{\hom(C_n,X),\hom(\partial_n,X)\}$; is it always a cochain complex? Can I always define some kind of ``equivalence'' between chain complexes and cochain complexes using a suitable contrav. functor? Can I conversely choose carefully $X$ in order not to have such an equivalence?