limit of chain complexes
1. Proposition
Let 
be a pointed category and 
the category of chain complexes.
Suppose there exists a diagram 
and all limits 
, then a limit of the chain complexes is given by
see: lim
2. Proof
2.1. special case
empty diagram, here the zero chain satisfies the requirements todo
2.2. construction
Consider a chain complex as diagram
Thus we get 
-shaped diagrams
obtained by postcomposition with n-th component functor of chain complexes.
By commutativity of chain maps, we obtain natural transformations from the boundary maps 
Hence taking the lim functor results in a diagram
It remains to show, that this is a valid chain complex.
Choose an indexed chain complex 
.
Then we get a commutative diagram
and by boundary condition a zero morphism
Note that 
makes the diagram commute, hence, since the morphism is induced by the universal property (cf. limit), we conclude, that 
is also the morphism in the bottom half
