  In this paper, we discuss certain circumstances in which the category of tame
functors inherits an abelian category structure with minimal resolutions and a
model category structure with minimal cofibrant replacements. We also present a
structure theorem for cofibrant objects in the category of tame functors
indexed by realizations of posets of dimension $1$ with values in the category
of chain complexes in an abelian category whose all objects are projectives.
Moreover, we introduce a general technique to generate indecomposable objects
in the abelian category of functors indexed by finite posets.
