Examples of sheaf on piles

I am looking for examples of sheaves on stacks (defined in Understanding the Definition of Sheaf on a Stack) that come from several varieties.

Let $ M $ to be a multiple then $ underline {M} $ is a stack associated with $ M $ (I can give details if need be).

We know when to call a battery card $ mathcal {D} rightarrow mathcal {C} $ a sheaf on the pile. assume $ mathcal {D} $ comes from a multiple $ M $ and $ mathcal {C} $ comes from a multiple $ N $ it is to say that we have $ underline {M} rightarrow underline {N} $.

Given a battery map $ F: underline {M} rightarrow underline {N} $we have a unique map $ F (id): M rightarrow N $ associated to $ F $.

What can we say about this card? All comments are welcomed.