For any two objects, define a chain complex to have components
(the collection of degree- maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements by
This defines a functor
By Definition 1 the 0-cycles in are collections of morphisms such that
This is precisely the condition for to be a chain map.
Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form
for a collection of maps . This are precisely the null homotopies.