The most essential data of the type CoherentSheaf in Macaulay2 is the representative module. For morphisms of sheaves, the data type requires a little more care, because even if $\mathcal{F}$ and $\mathcal{G}$ are sheaves represented by modules $M$ and $N$, respectively, a morphism of sheaves $\phi : \mathcal F \to \mathcal G$ is not necessarily the sheaf associated to a module map $\psi : M \to N$. Indeed, the best one can say is that $\phi$ is represented by some map $$\psi : M_{\geq d} \to N,$$ where $d$ is some truncation degree. This means that in Macaulay2, a morphism of sheaves is represented as a morphism from some truncation of the source representative to the target representative.
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As illustrated in the above example, the source and target are still represented by the sheaves $\mathcal F$ and $\mathcal G$. The key degree accesses the truncation degree needed to represent the map as a morphism of modules. To access the actual matrix representing the map, use matrix.
The source of this document is in Varieties/doc-maps.m2:147:0.