  Using the properties of the local Boltzmann weights of integrable
interaction-round-a-face (IRF or face) models we express local operators in
terms of generalized transfer matrices. This allows for the derivation of
discrete functional equations for the reduced density matrices in inhomogeneous
generalizations of these models. We apply these equations to study the density
matrices for IRF models of various solid-on-solid type and quantum chains of
non-Abelian $\bm{su(2)_3}$ or Fibonacci anyons. Similar as in the six vertex
model we find that reduced density matrices for a sequence of consecutive sites
can be 'factorized', i.e.\ expressed in terms of nearest-neighbour correlators
with coefficients which are independent of the model parameters. Explicit
expressions are provided for correlation functions on up to three neighbouring
sites.
