Category of chain complexes with chain maps

Definition

Of abelian groups

The category of chain complexes of abelian groups with chain maps is defined as follows:

Aspect	Name	Description (data)	Compatibility condition that must be satisfied
Object	Chain complex	The following data describe an object $C$: for each integer $n$, an abelian group $C_{n}$, and a group homomorphism $\partial _{n}:C_{n}\to C_{n-1}$	$\partial _{n-1}\circ \partial _{n}=0$ for all $n$.
Morphism	Chain map	A morphism $f$ between objects $C$ and $D$ gives, for each integer $n$, a group homomorphism $f_{n}:C_{n}\to D_{n}$	$\partial _{n}\circ f_{n}=f_{n-1}\circ \partial _{n}$, where the $\partial _{n}$ on the left is in $D$ and $\partial _{n}$ on the right is in $D$.
Composition of morphisms		For a morphism $f:C\to D$ and a morphism $g:D\to E$, we define the composite morphism $h=g\circ f$ by $h_{n}=g_{n}\circ f_{n}$.

Of modules over a ring

Suppose $R$ is a commutative unital ring. The category of chain complexes of $R$-modules with chain maps is defined as follows:

Aspect	Name	Description (data)	Compatibility condition that must be satisfied
Object	Chain complex	The following data describe an object $C$: for each integer $n$, a $R$-module $C_{n}$, and a $R$-module homomorphism $\partial _{n}:C_{n}\to C_{n-1}$	$\partial _{n-1}\circ \partial _{n}=0$ for all $n$.
Morphism	Chain map	A morphism $f$ between objects $C$ and $D$ gives, for each integer $n$, a $R$-module homomorphism $f_{n}:C_{n}\to D_{n}$	$\partial _{n}\circ f_{n}=f_{n-1}\circ \partial _{n}$, where the $\partial _{n}$ on the left is in $D$ and $\partial _{n}$ on the right is in $D$.
Composition of morphisms	For a morphism $f:C\to D$ and a morphism $g:D\to E$, we define the composite morphism $h=g\circ f$ by $h_{n}=g_{n}\circ f_{n}$.