Igor wrote:

>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.

Yes; more generally we spent a lot of time talking about posets, where we can only have a morphism \\(f: x \to y\\) and a morphism \\(g: y \to x\\) if \\(x = y\\).

> 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?

It sounds like you're taking the visual metaphors very seriously here: the concept of 'sideways' doesn't mean much in a general category or even a poset. However, here is what's true: if we have two functors \\(F,G: \mathcal{C} \to \mathcal{D}\\) and \\(\mathcal{D}\\) is actually a poset, we can only have a natural transformation \\(\alpha: F \Rightarrow G\\) if

\[ F(x) \le G(x) \]

for all \\(x \in \mathrm{Ob}(\mathcal{C})\\).

This follows immediately from the definitions.

But posets are very special categories.

> What are we getting from following the topology of this category, except for constraints on possible natural transformations?

The word 'topology' doesn't apply here: there is no topology here.

>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.

Yes; more generally we spent a lot of time talking about posets, where we can only have a morphism \\(f: x \to y\\) and a morphism \\(g: y \to x\\) if \\(x = y\\).

> 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?

It sounds like you're taking the visual metaphors very seriously here: the concept of 'sideways' doesn't mean much in a general category or even a poset. However, here is what's true: if we have two functors \\(F,G: \mathcal{C} \to \mathcal{D}\\) and \\(\mathcal{D}\\) is actually a poset, we can only have a natural transformation \\(\alpha: F \Rightarrow G\\) if

\[ F(x) \le G(x) \]

for all \\(x \in \mathrm{Ob}(\mathcal{C})\\).

This follows immediately from the definitions.

But posets are very special categories.

> What are we getting from following the topology of this category, except for constraints on possible natural transformations?

The word 'topology' doesn't apply here: there is no topology here.