Also a more concrete question, following the previous one:

Some of the categories we've seen form directed lattices with a single source and sink (initial and terminal objects), like a power set of some set S with \\(\subseteq\\) as morphisms.

Is it true that if we have images F and G in such a category, the F can only be naturally transformed into G in a limited number of ways - it may slide upwards (if morphisms allow) and/or expand/shrink sideways and upwards? What are we getting from following the topology of this category, except for constraints on possible natural transformations?

Other categories, resembling manifolds, have a property of having non-empty two-way sets of morphisms for all objects and their "neighborhoods". In such categories possible natural transformations greatly expand their repertoire, so there are "translations", "rotations", "scaling", non-rigid deformations, everything having topological taste, because natural transformations require all these to happen continuously along morphisms. Is this mental picture right, can it somehow be expanded, or, conversely, constrained?

**EDIT:** writing this I realized that at least the category **FinSet** without the empty set has a codiscrete (indiscrete) "topology", where each object is connected to another. So if there is a nice way (preserving object relationships) to morph F into G, it is a natural transformation. Btw, from what I see at the moment, a natural transfomation between F and G always exists in this category, don't know whether this is true or not.

Some of the categories we've seen form directed lattices with a single source and sink (initial and terminal objects), like a power set of some set S with \\(\subseteq\\) as morphisms.

Is it true that if we have images F and G in such a category, the F can only be naturally transformed into G in a limited number of ways - it may slide upwards (if morphisms allow) and/or expand/shrink sideways and upwards? What are we getting from following the topology of this category, except for constraints on possible natural transformations?

Other categories, resembling manifolds, have a property of having non-empty two-way sets of morphisms for all objects and their "neighborhoods". In such categories possible natural transformations greatly expand their repertoire, so there are "translations", "rotations", "scaling", non-rigid deformations, everything having topological taste, because natural transformations require all these to happen continuously along morphisms. Is this mental picture right, can it somehow be expanded, or, conversely, constrained?

**EDIT:** writing this I realized that at least the category **FinSet** without the empty set has a codiscrete (indiscrete) "topology", where each object is connected to another. So if there is a nice way (preserving object relationships) to morph F into G, it is a natural transformation. Btw, from what I see at the moment, a natural transfomation between F and G always exists in this category, don't know whether this is true or not.