Let $X,Y$ be topological spaces, $\mathsf{C},\mathsf{D}$ categories, $I$ the interval, $\mathbb{2}$ the 2-object category with one nontrivial morphism, and $\mathsf{I}$ the 2-object category with two nontrivial inverse morphisms.
Then we say that two functors $F,G : \mathsf{C} \rightrightarrows \mathsf{D}$ are naturally isomorphic if there is a functor $H : \mathsf{C} \times \mathsf{I} \to \mathsf{D}$ so that, for the inclusions $i_0,i_1 : \mathsf{C} \hookrightarrow \mathsf{C} \times \mathsf{I}$, $F$ and $G$ factor through $H$ as follows: $H i_0 = F$ and $Hi_1 = G$.
The classical analogy is that natural isomorphisms are to functors are homotopies are to functions; indeed, replace $\mathsf{C}$ with $X$, $\mathsf{D}$ with $Y$, and $\mathsf{I}$ with $I$, and the above definition in $\mathsf{Top}$ is precisely the definition of a homotopy $H$ between $F$ and $G$.
This makes me wonder: what is the topological analogue to a noninvertible natural transformation?
A natural transformation can be defined by the above factoring, only we replace $\mathsf{I}$ with $\mathbb{2}$. This seems to suggest the idea of a "directed interval" we replace $I$ with, relaxing the requirement of a homotopy to only require one direction. But since one-directional homotopies imply bidirectional homotopies (simply "walk backwards" along the one-directional homotopy), this is not the correct notion. I have other thoughts on how to enforce extra structure, but I was wondering if more experienced category theorists or students know of a canonical notion which works-- or even better, why there isn't one!
EDIT: A pospace seems promising, is there a standard discussion someone can direct me to?