A limit over a diagram of this form is called a sequential limit/directed limit. If all connecting morphisms are epimorphism then we sometimes say that the tower is a cofiltration? (though the same term may be sometimes applied more generally for more general index categories, or more restrictively when there is an object in the ambient category such that all are its quotients).
Different contexts lead to different notions of morphism of towers, so it is important to consider what category of towers is appropriate for any given use of these objects. In addition to Postnikov towers, and related uses in decomposing homotopy types, towers also occur as a simple type of pro-object in a category. In that situation the morphisms considered between towers are usually pro-morphisms.
The tower diagram shape is evidently a small cofiltered category. As such it makes sense to consider pro-morphisms between tower diagrams:
For any small category , the full subcategory
(e.g. Blanc 96, def. 2.5)
For a tower diagram, we write
for its formal cofiltered limit, i.e.
Notice that generally we have that morphisms between formal cofiltered limits (pro-objects) of shapes and , respectively, are represented as cofiltered limits of systems of morphisms indeced on a single cofiltered category equipped with final functors to and , respectively. (See at ind-object this prop.).
The following says that in the case that both shapes are towers, then also may always be chosen to be of tower shape:
For and two tower diagrams in , then every morphism
between their formal cofiltered limits (every pro-morphism between the diagrams) in is represented by component morphisms
making commuting diagrams
i.e. by a natural transformation between