I am trying to prove the following presumably easy fact: if $B$ is a category with small hom-sets and $C$ is a small category, then $B^{C}$ has small hom-sets. I am assuming the standard Z-F axioms with the additional assumption of a universe, which is a set $U$ satisfying some standard closure conditions. A set $x$ is $\textit{small}\Leftrightarrow x\in U$.
The proof seems straightforward enough, but the very last step is giving me trouble. If $F,G:C\rightarrow B$ are fixed objects in $B^{C}$ and $\tau :F\overset{.}{\rightarrow}G$ is an arrow in $B^{C}$, then $\tau$ is a function:$C\rightarrow \bigcup_{c\in C}$ [ $Fc$, $Gc$]. Since $C$ is small, and the union of the small sets on $RHS$ is small, the result follows.
But I do not see why the union is small. It is certainly true that we can get a set $\Im =\left \{[Fc,Gc ]:c\in C\right \}$, using the limited comprehension axiom, since each hom-set is in $U$. The union then also exists as a set, which is just $\bigcup_{c\in C}[Fc,Gc]$.
My question is: $\Im $ is a set but why is it in $U$?