Category of chain complexes with chain maps

Definition

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 $h_{n} = g_{n} \circ f_{n}$ .

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 $h_{n} = g_{n} \circ f_{n}$ .