Limits/Colimits as representing objects and size of functor category Let $J,C$ be categories and $T\in C$, we define the functor $\Delta_J(T):J \to C, j \to T $ to be the constant $T$-valued $J$-diagram.
Let $F:J \to C$ be a functor, my professor defined the limit of $F$ to be a representation of the functor $Hom_{Fun(J,C)}(\Delta_J(-),F)$. Because representing objects are unique up to isomorphism, we instantly get uniqueness of limits.
However this only makes sense to me when $Fun(J,C)$, ( the class of functors from $J$ to $C$) is a locally small category, which seems to be the case when $J$ is small and $C$ is locally small, but I have no reference.
We have only been working with locally small categories, so my questions are:
Is $Fun(J,C)$ locally small, when $J$ is small and $C$ is locally small?
How to make sense of the other definitions of limits, where $J$ is not required to be small? How is uniqueness proven?
 A: 
Is $Fun(J, C)$ locally small when $J$ is small and $C$ is locally small?

Yes. A natural transformation between functors $F$ and $G$ is just an element of $\prod\limits_{A \in J} Hom(FA, GA)$ satisfying the naturality condition. So the class of natural transformations from $F$ to $G$ is a subclass of the set $\prod\limits_{A \in J} Hom(FA, GA)$, hence is a set.

How to make sense of other definitions of limits when $J$ is not required to be small?

Let’s start by assuming $J$ is small and $C$ is locally small. Note that representations of a functor $H : C^{op} \to Set$ are in natural bijection with objects $A$, together with some $a \in HA$, such that for all $b \in HB$ there is a unique $f : Hom(B, A)$ such that $b = H(f)(a)$. This follows from the Yoneda lemma.
In this case, a representation of the functor $Hom(\Delta_J(-), F)$ corresponds to an object $A \in C$, together with some natural transformation $\pi : Hom(\Delta_J(A), F)$ such that for all $b \in Hom(\Delta_J(B), F)$, there exists a unique $f : B \to A$ such that $b = \pi \circ \Delta_J(f)$.
It turns out that we can now define the limit to be such an object $A$ together with such a natural transformation $\pi$. This definition works even when $J$ and $C$ are not assumed locally small. Furthermore, it’s fairly straightforward to prove that limits are unique. For if we had two limits $(A, \pi)$ and $(B, p)$, we could construct back-and-forth maps $f : Hom(A, B)$ and $g : Hom(B, A)$ which are inverse isomorphisms and which preserve the $\pi/p$ structure.
Note that phrasing this will involve quantifying over a collection of proper classes. So even stating the definition of limits over large diagrams requires a meta theory like NBG (a class theory which is a “conservative extension” of ZFC - anything provable about sets in NGB is also provable in ZFC) or MK (which is much, much stronger than ZFC). An alternate tactic is to use Grothendieck universes, which allows you to keep using set theory but requires the adoption of some very strong large cardinal axioms.
