When I first went from CT as an implementation detail of typed FP to studying CT on its own, I was very aggravated by this point: "where is there a functor that isn’t an endofunctor?"
Over time it has fallen away in two directions.
In one direction is that though you are always stuck in a category with types as objects and functions as arrows, that doesn't stop you from subdividing. This became more intuitive to me as I encountered more and more categories that are basically just Set with extra structure (just as the one we program in is basically Set with bottom). If you start putting extra structure on your arrows (that is something that carries around a function along with some extra proof-relevant structures, like say a way to show that some zero element in the domain under the function equals the zero element in the codomain and that multiplication is preserved under the function, now you have monoid-homomorphisms) then you end up in the general case with something that looks a lot like a binatural transformation of profunctors, which pretty cleanly encodes "exo"-functors between different sub-categories of the category with all types as objects and functions as arrows.
In the other direction is realizing that even imprecisely, there's quantifiable value in observing functors to and from the category of types and functions, say one between whiteboard diagrams and code, or between a specific problem domain and code, etc etc. If you have some other space with composable relationships that you want to preserve when you write code to correspond to it, or you have some other representation you want to produce based off some given codebase while preserving the structure of the code taken as input, you can gain a ton of conceptual leverage out of identifying a functor.
Over time it has fallen away in two directions.
In one direction is that though you are always stuck in a category with types as objects and functions as arrows, that doesn't stop you from subdividing. This became more intuitive to me as I encountered more and more categories that are basically just Set with extra structure (just as the one we program in is basically Set with bottom). If you start putting extra structure on your arrows (that is something that carries around a function along with some extra proof-relevant structures, like say a way to show that some zero element in the domain under the function equals the zero element in the codomain and that multiplication is preserved under the function, now you have monoid-homomorphisms) then you end up in the general case with something that looks a lot like a binatural transformation of profunctors, which pretty cleanly encodes "exo"-functors between different sub-categories of the category with all types as objects and functions as arrows.
In the other direction is realizing that even imprecisely, there's quantifiable value in observing functors to and from the category of types and functions, say one between whiteboard diagrams and code, or between a specific problem domain and code, etc etc. If you have some other space with composable relationships that you want to preserve when you write code to correspond to it, or you have some other representation you want to produce based off some given codebase while preserving the structure of the code taken as input, you can gain a ton of conceptual leverage out of identifying a functor.