\begin{definition}[Definition:Homomorphism of Complexes]
Let $\struct {R, +, \cdot}$ be a ring.
Let:
:$M: \quad \cdots \longrightarrow M_i \stackrel {d_i} {\longrightarrow} M_{i + 1} \stackrel {d_{i + 1} } {\longrightarrow} M_{i + 2} \stackrel {d_{i + 2} } {\longrightarrow} \cdots$
and
:$N: \quad \cdots \longrightarrow N_i \stackrel {d'_i} {\longrightarrow} N_{i + 1} \stackrel {d'_{i + 1} } {\longrightarrow} N_{i + 2} \stackrel {d'_{i + 2} } {\longrightarrow} \cdots$
be two differential complexes of $R$-modules.
Let $\phi = \set {\phi_i: i \in \Z}$ be a family of module homomorphisms $\phi_i: M_i \to N_i$.
Then $\phi$ is a '''homomorphism of complexes''' {{iff}} for each $i \in \Z$:
:$\phi_{i + 1} \circ d_i = \phi_i \circ d'_i$
That is, for each $i \in \Z$ we have a commutative diagram:
::$\begin{xy}\xymatrix@L+2mu@+1em {
 M_i \ar[r]^*{d_i}
     \ar[d]^*{\phi_i} &
M_{i+1} \ar[d]^*{\phi_{i+1}} \\
 N_i \ar[r]^*{d'_i} &
 N_{i+1} } \end{xy}$
\end{definition}