In the derived category (i.e. the localisation of the homotopy category of chain complexes at all quasi-isomorphisms) $D(\mathcal{A})$ we know that morphisms can be thought of as roofs, or spans:
This is confirmed in, e.g. Dimca's Sheaves in Topology.
However, when we localise a category (at a right-multiplicative set, say) the morphisms are instead cospans (or coroofs, I guess?):
This is confirmed in, e.g. Schapira's Categories and homological algebra: an introduction to derived categories.
Question: why do we have this incongruence? Is it because Schapira considers a right-multiplicative set instead of a left one? Or is there something going on instead with the fact that Dimca uses cohomologically-graded cochain complexes (i.e. the differential increases degree)? Or is there something much deeper here, i.e. an actual fundamental difference.
When a category $\mathcal{C}$ admits a calculus of right fractions with respect to $\mathcal{S}$, the localization $\mathcal{C} [\mathcal{S}^{-1}]$ may be described in terms of equivalence classes of spans. When it admits a calculus of left fractions, the localization may be described via equivalence classes of co-spans. When it admits both, the resulting constructions coincide. For details, see https://ncatlab.org/nlab/show/calculus+of+fractions
The homotopy category $K (\mathcal{A})$ with respect to quasi-isomorphisms admits both a calculus of right and left fractions.
Homological and cohomological situation is essentially the same, as one only changes the numbering.