Some questions on Grothendieck topologies

Question

Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology on it, that is an assignment $J(C)$, for any $C \in \mathcal{C}$ assigning to any object a collection of sieves (think of sieves as suitable sinks). Then $J(C)$ is a suitable collection of sieves, i.e. a suitable collection of specific collections of morphisms of $\mathcal{C}$. Now, as $\mathcal{C}$ is a small category, it results that any sieve forms a set of morphisms. What about $J(C)$? Is it again a set or is it a class? In addition, let us consider the family $\mathfrak{I}:=\{J(C): \, C \in \mathcal{C}\}$. Does it form a set or a class? And finally, if $\mathfrak{T}$ agrees with the collection of all the possible collections of sinks in $\mathcal{C}$, what can we say about $\mathfrak{T}$?