You are correct that the first notion of U-category corresponds more
closely to non-Grothendieck-universe-based treatments, e.g. using NBG
or MK set-class theory. To be precise, if U is a universe, defining
"set" to mean "element of U" and "class" to mean "subset of U" gives a
model of MK set-class theory (and hence also NBG, which is weaker than
MK). A comparison of the relationships between different set-theoretic
treatments of large categories can be found in my expository paper
"Set theory for category theory."
Here is an example of one theorem that can (maybe) tell the difference
between the two notions of U-locally-small categories. Let C be a
U-category whose hom-sets are in U (i.e. a "U-locally-small category"
by your second definition). Then C has a Yoneda embedding C →
[Cº,Set] where Set is the U-category of U-small sets. Note that
[Cº,Set] is only a U-category by your second definition (i.e. a
U⁺-small category). We say that C is lex-total if this Yoneda
embedding has a left adjoint which preserves finite limits. It is a
theorem of Freyd, which can be found in Ross Street's paper "Notions
of topos," that if C is lex-total and also U-locally-small according
to your first definition (its set of objects is a subset of U), then C
is a Grothendieck topos (i.e. the category of U-small sheaves on some
U-small site). The converse is not hard to prove, so this gives a
characterization of Grothendieck toposes. As far as I know, it is
unknown whether there can be lex-total U-categories with very large
object sets that are not Grothendieck toposes.
I would personally be inclined to use your second definition of
"U-locally small," because as you say it matches your preferred
definition of large category relative to U (which I would prefer to
just call a "U⁺-small category", since its definition makes no reference
to U), and also because the term "U-locally small" sounds as if it
only imposes a smallness condition locally. Street uses "moderate"
for a category with at most a U-small set of isomorphism classes of
objects, so if one wants to state a theorem (such as the above) about
U-locally-small categories according to your first definition, one can
instead say "U-locally-small and U-moderate."
I predict that someone such as Steve Lack or Mike Shulman will tell you about the existence of (co)limits in Mon, and they'll do it better than I would, so instead I'll address a question in the last paragraph: do $M(0)$, $M(1)$ and $M(0 \to 1)$ tell you much about the rest of $M$? The answer is basically no.
To see this -- and to understand monads -- it's helpful to observe that if $M$ is regarded as an algebraic theory then $M(n)$ is the set of words in $n$ letters, or equivalently $n$-ary operations in the theory. For example, if $M$ is the monad for groups then $M(n)$ is the set of words-in-the-group-theory-sense in $n$ letters, which are the same as the $n$-ary operations in the theory of groups. For example, $x^3 y^2 x^{-1}$ is a typical word in two letters, and $(x, y) \mapsto x^3 y^2 x^{-1}$ is a typical binary operation (way of turning a pair of elements of a group into a single element). Similarly, if $M$ is the monad for rings then $M(n)$ is the set of polynomials over $\mathbb{Z}$ in $n$ variables, which are the $n$-ary operations in the theory of rings.
(Personally I'm happy to think of any monad on Set as an algebraic theory, although others prefer a more restrictive definition of theory. Anyway, this point of view doesn't matter in what follows.)
In particular, $M(0)$ is the set of nullary operations, or constants, in the theory, and $M(1)$ is the set of unary operations. Now let's consider on the one hand the identity monad $I$, and on the other the monad $M$ corresponding to the theory generated by a single binary operation $\cdot$ subject to the equation $x\cdot x = x$. (Or if you want a more familiar but more complicated example, take $M$ to be the monad for [not necessarily bounded] lattices or semilattices; the point is that $x \vee x = x$.) We then have $I(0) = 0 = M(0)$ and $I(1) = 1 = M(1)$, hence also
$$
I(0 \to 1) = (0 \to 1) = M(0 \to 1).
$$
But $I$ and $M$ are very different monads.
On the positive side, I think you might be interested in the following result; it seems in the spirit of your question. Suppose there exists an $M$-algebra with more than one element. Then:
- $M$ reflects isomorphisms
- $M$ is faithful
- the unit $\eta$ of $M$ is monic (i.e. the free $M$-algebra on a set of generators contains the generators as a subset -- it doesn't identify any of them).
What's more, all but two monads on Set have this property that there exists an algebra with more than one element. One of the exceptions is the monad $M$ with $M(A) = 1$ for all sets $A$; it's the theory generated by a single constant $e$ and the equation $x = e$. The other is the monad $M$ with $M(A) = 1$ for all nonempty sets $A$ and $M(0) = 0$; that's the theory generated by no operations and the equation $x = y$.
Edit David asked where a proof of this "positive" result could be found. I learned it from a Cambridge Part III problem sheet by Peter Johnstone, and it's probably out there somewhere in the literature (e.g. maybe in Johnstone's Sketches of an Elephant). But since consulting the literature means getting up from the couch, I'll type out a proof instead.
So, let $M = (M, \eta, \mu)$ be a monad on Set such that there exists at least one $M$-algebra with more than one element. Write $F: \mathrm{Set} \to \mathrm{Set}^M$ for the free algebra functor, and $U: \mathrm{Set}^M \to \mathrm{Set}$ for the underlying set functor; thus, $M = UF$.
First we show that $\eta: id \to M$ is monic. This means for each set $A$ and each $a, b \in A$ with $a \neq b$, we have $\eta_A(a) \neq \eta_A(b)$. By the universal property of $\eta_A$, this is equivalent to the assertion that for some $M$-algebra $X$ and map $f: A \to U(X)$ of sets, we have $f(a) \neq f(b)$. (I'd like to draw a commutative triangle to illustrate this.) Choose any $M$-algebra $X$ with more than one element: then such an $f$ can indeed be constructed.
Next we show that $F$ is faithful. It will follow that $M = UF$ is faithful, since $U$ (being monadic) is also faithful. Faithfulness of $F$ is in fact equivalent to each $\eta_A$ being monic, by general properties of adjunctions; I think that's in Categories for the Working Mathematician. Anyway, the proof is that if $f, g: A \to B$ are maps of sets with $Ff = Fg$ then $\overline{Ff} = \overline{Fg}$ (where the bar indicates transpose); but $\overline{Ff} = \eta_B \circ f: A \to UF(B)$ and similarly for $g$, and $\eta_B$ is monic, so $f = g$.
Finally we show that $F$ reflects isos. It will follows that $M = UF$ reflects isos, since $U$ (being monadic) also reflects isos. A faithful functor always reflects both epis and monos, and any map in Set that's both epi and mono is an iso. Hence any faithful functor out of Set reflects isos, and the result follows from the previous part.
Best Answer
In response to your second question, this is not precisely what you're looking for, but here is one quote by Yuri Manin along the same lines:
All the other vehicles of mathematical rigor are secondary [to definitions], even that of rigorous proof.
Manin makes this remark in an essay, entitled "Interrelations between Mathematics and Physics" that contains more memorable phrases, such as the final line of this extended quote: