  We formalize the way in which one can think about cluster algebras of
infinite rank by showing that every rooted cluster algebra of infinite rank can
be written as a colimit of rooted cluster algebras of finite rank. Relying on
the proof of the posivity conjecture for skew-symmetric cluster algebras (of
finite rank) by Lee and Schiffler, it follows as a direct consequence that the
positivity conjecture holds for cluster algebras of infinite rank. Furthermore,
we give a sufficient and necessary condition for a ring homomorphism between
cluster algebras to give rise to a rooted cluster morphism without
specializations. Assem, Dupont and Schiffler proposed the problem of a
classification of ideal rooted cluster morphisms. We provide a partial solution
by showing that every rooted cluster morphism without specializations is ideal,
but in general rooted cluster morphisms are not ideal.
