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Some of what motivated the questions

A common definition of Functors is that a functor $F : \mathbf{A} \to \mathbf{B}$ from category $\mathbf{A}$ to category $\mathbf{B}$ consists of:

  1. A map $F_{objects}: ob(\mathbf{A}) \to ob(\mathbf{B})$ from the collection of objects of $\mathbf{A}$ to the collection of objects of $\mathbf{B}$
  2. A map $F_{arrows} : ar(\mathbf{A}) \to ar(\mathbf{B})$ from the collection of arrows of $\mathbf{A}$ to the collection of arrows of $\mathbf{B}$

But we can also define functors as arrows in suitable categories which act on generalized elements in a way such to induce maps. Let $\mathbf{1}$ be the category with precisely one object with only identity morphism and $\mathbf{2}$ the category with precisely distinct terminal and initial object in suitable $\mathbf{K}$-enriched category $\mathbf{C}$ for all categories $\mathbf{A}$,$\mathbf{B}$ in $\mathbf{C}$.

We may take arrow $F:\mathbf{A}\to\mathbf{B}$ to be a functor

Then since $hom_\mathbf{C}(\mathbf{1},\mathbf{A})\cong ob(\mathbf{A})$, we may for compose $F$ with each functor $a:\mathbf{1}\to\mathbf{A}$ to get $F\circ a:\mathbf{1}\to\mathbf{B}$ which is an object of $\mathbf{B}$ then we construct a map elementwise in $\mathbf{K}$ from $ob(\mathbf{A})$ to $ob(\mathbf{B})$.

As $hom_\mathbf{C}(\mathbf{2},\mathbf{A})\cong ar(\mathbf{A})$ we may perform an analogous construction to make a map from $ar(\mathbf{A})$ to $ar(\mathbf{B})$ from $F$

Making it look like the definition as arrows (and thus maps of generalized elements) subsumes the definition as pairs of maps.

But the definition as arrows was made while keeping an eye on the definition as maps, suggesting that the definition as certain types of arrows should be subsumed by the definition as pairs of maps.

The questions

How do we prove that all the functors defined as pairs of maps can be created from arrows in the correct categories? What are the minimum properties needed for a category to have maps that induce all functors? (Possible restricted to between categories of specific sizes, but if a size constraints need to be chosen, there would hopefully be analogous constructions for any given size)

Can we show that an arrow $F$ in a suitable category (not necessarily $\mathbf{Cat}$) is defined entirely by the induced maps on the objects and arrows in the category over which the suitable category is enriched over. What properties do the suitable categories need to have to meet this condition?

How may we show that the mappings of the different generalized objects are consistent? For a simple example give $a$ is object in $\mathbf{A}$ and arrow $g$ in $\mathbf{A}$ with source (or target) $a$, how can we be sure that if $a$ is mapped to $b$ by $F$ and $g$ is mapped to $h$ by $F$ that the source (or target) of $h$ is $b$? Does this impose any additional conditions? I would hypothesize that it does not

Can similar things be shown with specific types of functors? (i.e. Can the rules for a monoidal functor be shown to be the same as being arrows in some sort of suitable "category of monoidal categories")

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This is all eating its own tail a bit. For instance to know that $hom_C(1,A)=Ob_A$, you already need to know what a functor is! But it's true in every category that a morphism can be described in terms of the mapping it induces on generalized elements. This is the intuitive content of the Yoneda lemma.

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