  We study the structure of the category of representations of $\mathbf{FA}$,
the category of finite sets and all maps, mostly working over a field of
characteristic zero. This category is not semi-simple and exhibits interesting
features.
  We first construct the simple representations, recovering the classification
given by Wiltshire-Gordon. The construction given here also yields explicit
descriptions of the indecomposable projectives.
  These results are used to give a convenient set of projective generators of
the category of representations of $\mathbf{FA}$ and hence a Morita equivalence
result. This is used to explain how to calculate the multiplicities of the
composition factors of an arbitrary object, based only on its underlying
$\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite
sets and bijections.
  This is applied to show how to calculate the morphism spaces between
projectives in our chosen set of generators, as well as for a closely related
family of objects (the significance of which can be shown by relative
nonhomogeneous Koszul duality theory).
