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I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me:

Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\mathcal{C}$ into a small category $\mathcal{D}$.

Then the object part of the functor, $F_{obj}$, associates with each object $c \in \mathcal{Ob(C)}$ an object $Fc \in \mathcal{Ob(D)}$, making it a function $F_{obj}:\mathcal{Ob(C)}\to \mathcal{Ob(D)}$ (with both the domain and codomain being sets since $\mathcal{C}$ and $\mathcal{D}$ are small categories).

Likewise, the morphism part of the functor, $F_{mor}$, associates with each morphism $f \in \operatorname{Hom}_{\mathcal{C}}(c, c')$ a morphism $Ff \in \operatorname{Hom}_{\mathcal{D}}(Fc, Fc')$, making $F_{mor}$ a family of functions $\{F_i: \operatorname{Hom}_{\mathcal{C}}(c,c')\to\operatorname{Hom}_{\mathcal{D}}(Fc,Fc')\ |\ (c,c')=i \in \mathcal{C}\times\mathcal{C} \}$.

Hence, ignoring the foundational problem of defining "functions" between proper classes as opposed to sets (i.e. the case when either $\mathcal{C}$ or $\mathcal{D}$ is not small), functors can be thought of as actual (families of) functions, not just arbitrary morphisms between categories.

morphism between categories - consider a category $\mathscr{C}$, for which the objects $\mathcal{C} \in\mathcal{Ob}(\mathscr{C})$ are themselves categories. Then the morphisms of the category $\mathscr{C}$ are "morphisms between categories".

Question: Given a category $\mathscr{C}$ of the type described above, are its morphisms always functors?

I believe not, since the structure described above for functors seems to be more than is strictly necessary to satisfy the category axioms (associativity, identity, closure under composition).

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    $\begingroup$ Functors are functions which satisfy the properties you listed. Although we usually think of categories as having objects and morphisms, we can also just think of them as having a set of morphisms with a partial binary operation defined on them satisfying some properties. Then functors are functions respecting that binary operation. "In fact functors have more structure than is strictly necessary to be a morphism between two categories, making them a distinct concept from morphism." -- I don't know what this means. Most people define a morphism of categories to be a functor. $\endgroup$ – Kyle Ferendo Jul 24 '16 at 20:25
  • $\begingroup$ @KyleFerendo Thank you for your comments -- I have tried to clarify the latter part of the question as well as what I meant by "morphisms between categories". I didn't realize when writing the question that it might be unclear to others. $\endgroup$ – Chill2Macht Jul 24 '16 at 20:39
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    $\begingroup$ We can use any sort of wild system for defining "morphisms between categories" if you just require that they be morphisms in some category in which the objects are nominally small categories. For instance, I could say, "Let $C$ be the category with objects small categories and exactly one morphism between any two objects." But when defining morphisms, they should respect the structure we care about on the objects, so that they can be used to gain information about the objects (as my example fails to do). Functors are the right way to do this for categories. $\endgroup$ – Kyle Ferendo Jul 24 '16 at 20:44
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    $\begingroup$ However, there are other "categories of small categories" that are actually useful to study. We could define morphisms more strictly, to be left adjoint functors. Or we could define morphisms to be correspondences; that is, a functor $A^\mathrm{op}\times B\to \mathrm{Set}$ could be a morphism $A\to B$. $\endgroup$ – Kyle Ferendo Jul 24 '16 at 20:45
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    $\begingroup$ In the framework described in your question, a functor is a pair of functions, the ones you called $F_{obj}$ and $F_{mor}$. $\endgroup$ – Andreas Blass Jul 24 '16 at 21:42
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For the sake of having an answer to this question for future readers, I'm copying over my comments:

Functors are functions which satisfy the properties you listed. Although we usually think of categories as having objects and morphisms, we can also just think of them as having a set of morphisms with a partial binary operation defined on them satisfying some properties. Then functors are functions respecting that (partial) binary operation.

If we call a "morphism of categories" a morphism in any category in which the objects are nominally categories, then a "morphism of categories" can be anything -- and probably won't actually have anything at all to do with categories. For instance, we could define a category in which the categories are small categories and hom-sets always contain a single morphism. This definition does not actually assist us in studying categories, which is the point of defining a category of some type of object: to use the morphisms in that category to learn more about those objects. But for that to happen, the morphisms have to respect some of the structure of the objects in question.

When we're dealing with categories, functors are pretty much the way to go, but there are a few other possibilities we might use. For instance, there is a way of defining composition for correspondences of categories. A correspondence $C\to D$ is a functor $C^\mathrm{op}\times D\to \mathbf{Set}$.

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