  In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy
invariant concepts in the category of chain complexes. Our arguments were based
on the fact that strongly homotopy algebras are algebras over minimal cofibrant
operads and on the principle that algebras over cofibrant operads are homotopy
invariant. In our approach, algebraic models for colored operads describing
diagrams of homomorphisms played an important role.
  The aim of this paper is to give an explicit description of these models. A
possible application is an appropriate formulation of the `ideal' homological
perturbation lemma for chain complexes with algebraic structures. Our results
also provide a conceptual approach to `homotopies through homomorphism' for
strongly homotopy algebras. We also argue that strongly homotopy algebras form
a honest (not only weak Kan) category.
  The paper is a continuation of our program to translate the famous book "M.
Boardman, R. Vogt: Homotopy Invariant Algebraic Structures on Topological
Spaces" to algebra.
