  In \cite{CompTheo} we studied the indeterminacy of the value of a derived
functor at an object using different definitions of a derived functor and
different types of fibrant replacement. In the present work we focus on derived
or homotopy limits, which of course depend on the model structure of the
diagram category under consideration. The latter is not necessarily unique,
which is an additional source of indeterminacy. In the case of homotopy
pullbacks, we introduce the concept of full homotopy pullback by identifying
the homotopy pullbacks associated with three different model structures of the
category of cospan diagrams, thus increasing the number of canonical
representatives. Finally, we define generalized representatives or models of
homotopy limits and full homotopy pullbacks. The concept of model is a unifying
approach that includes the homotopy pullback used in \cite{JL} and the homotopy
fiber square defined in \cite{Hir} in right proper model categories. Properties
of the latter are generalized to models in any model category.
