Can you "construct" a category from another category? It's perfectly possible that I have no idea what I'm talking about here, as I know little about category theory, but my understanding is that a category is a space of some objects equipped with some "arrow" (functions in a loose sense) on those objects. Since it seems that these categories are defined typically quite independently from each other, and in a standard undergrad curriculum you might see the "bottom-up" construction of certain spaces from more elementary spaces, I am wondering if you can somehow "construct" one category from the other (and so in a sense they are not independent).
As an example, it is my understanding that you can (in principle) construct much of mathematics starting from nothing more than the ZFC axioms of set theory (the first few lectures in this series does this nicely; in particular, he constructs topological spaces.). This makes it look like (to my naive eyes) you can construct Top from Set. However, reading on the Axiom of Choice, we know that it "fails" on Top, but "works" in Set, so presumably there is some contradiction. See for example, this answer on Math Overflow (incidentally, I don't totally understand this answer; topologists use the Axiom of Choice in the form of Tychonoff's Theorem or Zorn's Lemma often by my understanding, yet the category they work in forbids it?).
Perhaps my misunderstanding lies in some difference between the category Set and ZFC set theory, or the space of topological objects that topologists work in and Top?
Edit: the comments gave some examples of categories constructed from other categories, a trivial one being the opposite category. Refining my question, I am wondering if you can specifically construct Top from Set, and if construction of objects (ignoring category theory) from ZFC implies that you can construct the category of those objects from Set. Also, is the axiom of choice (as in the linked MO question) problematic, because it "fails" in some categories but not others?
 A: The statement "Choice fails in Top" has a very specific technical meaning, and it's not "topologists can't use Choice." Rather, the point is this: many statements about sets can be reformulated in terms of maps alone. Such reformulations then make sense in the context of arbitrary categories - although they may only hold in some.
Here are a couple examples:

*

*Take the theorem "There is a set $x$ such that for any set $y$ there is exactly one function from $x$ to $y$" (namely $x=\emptyset$). In categorical terms, this is the statement that Set has an initial object. Some categories have initial objects while others don't, so we see here an example of a set-theoretic principle holding in some but not all categories.


*A more interesting example is provided by the Cantor-Bernstein theorem. Classically this says that whenever we have a pair of injections from each of two sets to the other, there is a bijetion between those two sets; its categorical formulation is "Given monomorphisms $f:A\rightarrow B$ and $g:B\rightarrow A$, there is an isomorphism $h:A\cong B$." This fails in Top for example since e.g. $(0,1)$ and $[0,1]$ with the usual topologies are not homeomorphic but continuously embed into each other. See e.g. here for a discussion of which categories satisfy this.
In the case of the axiom of choice, the relevant statement is "Surjections split," or "For every surjective $f:X\rightarrow Y$ there is a map $g:Y\rightarrow X$ such that $f\circ g=id_Y$." This can be reformulated in categorical terms (replacing "surjection" with "epimorphism"), and holds in some but not all categories. In particular, we have for example:

$\mathsf{ZFC}$ proves that choice holds in Set but fails in Top.


So there is no obstacle to Top being "buildable" from Set. This can in fact be done in a precise way: in the model-theoretic sense, Top is interpretable in Set. This is a rather tedious construction, and certainly more complicated than e.g. the construction of Set$^{op}$ from Set, but it's perfectly sensible.
A good first step is to familiarize yourself with the definition of "topological space" within the $\mathsf{ZFC}$ context. This will help motivate the categorical construction. Note that it's not quite true that we can recover the universe of sets $V$ from the category Set, but there is also plenty of information we do not lose in the passage $V\leadsto$ Set, and this is one such case.
A: A broad family of categories - aka algebraic structures - can be constructed from sets with a unified kind of operation.
There is a generalization of categories that are called "operads": here you prescribe not only arrows from $x$ to $y$, but also " multiarrows" from $x_1, \ldots, x_n$ to $y$. You can imagine these as corollas with root $y$ and leaves $x_1, \ldots, x_n$. If you then have $n$ new corollas with roots $x_1, \ldots, x_n$, you can compose them through the old corolla and get a new one based at $y$. Beside the details, this is a way of generalizing composition to multimaps.
An operad is now associated to a certain kind of structures; for example, there will be the operad $Ass$ that encodes the axioms for a monoid (an associative multiplication with unity). A functor of operads generalized exactly a function of categories: it sends objects to objects, multiarrows to multiarrows; it preserves identity and composition.
Also, given an ordinary category with a "tensor product" (like the product in set and the tensor product in vector spaces), it has a natural operad structure with multimaps given by
$$Mul(x_1, \ldots, x_n ; y) := Hom(  x_1 \otimes \ldots \otimes x_n, y) $$
It turns out that if you want to define the category of, say, all monoids, it coincides with the category of functors from $Ass$ to $Set$, where the latter is equipped with the operad structure given by the ordinary product of sets.
What is the advantage of such reformulation? We actually had to define an operad $Ass$ (say "the axioms") to have a well defined category $Fun(Ass,Set)$ of $Ass$-algebras (the algebras defined by the axioms). Now say you want to define the category of $\mathbb{C}$-algebras, i.e. associative rings over complex numbers. This is the same as a monoid where the ground set is a vector space, and the multiplication is bilinear; in other the words, the functors should be $Vec_{\mathbb{C}}$ valued. We get for free that $\mathbb{C}-Alg=Fun(Ass, Vec_{\mathbb{C}}) $.
In this way you get for free something: you separated the degree of freedom about the axioms, and about where the axioms take place. But remember: if you want to create something new, you always have to put new information somewhere. I also recommend you look up the ind-category construction.
