A map of complexes $f : C' \rightarrow D'$ is a sequence of maps $f_i : C'_i \rightarrow D'_{d'+i}$. The new map $g : C \rightarrow D$ is the sequence of maps $g_i : C_i \rightarrow D_{d+i}$ induced by the matrix of $f_i$.
One use for this function is to get the new map of chain complexes obtained by shifting the source or target of an existing chain map. For example, one can regard the differential on a complex can be regarded as a map of degree zero between shifted complexes.
i1 : R = ZZ/101[a,b,c];
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i2 : C = freeResolution coker vars R
1 3 3 1
o2 = R <-- R <-- R <-- R
0 1 2 3
o2 : Complex
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i3 : f = map(C[-1], C, dd^C, Degree => 0)
1 3
o3 = 1 : R <------------- R : 1
| a b c |
3 3
2 : R <-------------------- R : 2
{1} | -b -c 0 |
{1} | a 0 -c |
{1} | 0 a b |
3 1
3 : R <-------------- R : 3
{2} | c |
{2} | -b |
{2} | a |
o3 : ComplexMap
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i4 : assert isWellDefined f
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i5 : assert(degree f == 0)
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i6 : assert isCommutative f
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i7 : assert isComplexMorphism f
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i8 : assert not isComplexMorphism dd^C
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