  We formulate the concept of minimal fibration in the context of fibrations in
the model category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of
simplicial sets, for a small index category $\mathcal{C}$. When $\mathcal{C}$
is an $EI$-category satisfying some mild finiteness restrictions, we show that
every fibration of $\mathcal{C}$-diagrams admits a well-behaved minimal model.
As a consequence, we establish a classification theorem for fibrations in
$\mathbf{S}^\mathcal{C}$ over a constant diagram, generalizing the
classification theorem of Barratt, Gugenheim, and Moore for simplicial
fibrations.
