Really quite a silly question, but I'm confused about the notation around limits in category theory. Given a diagram of shape $I$, say $F$, I would write the limit of $F$ in some category as $\operatorname{lim}_{I}F$. But recently I saw someone write $\operatorname{lim}_{i \in I} F(i)$.
Something like $Hom_C(-,\operatorname{lim}_{i \in I}F(i)) \cong \operatorname{lim}_{i\in I} Hom_{C}(-,F(i))$ was stated ( the claim being that the Yoneda functor preserves limits).
Does this mean the same thing as $\operatorname{lim}_{I}F$? So in this case does this mean,
$Hom_C(-,\operatorname{lim}_{I}F) \cong \operatorname{lim}_{I} Hom_{C}(-,F)$?
Since $F(i)$s are objects and not functors, I'm not sure what else this could mean.Thanks!