Let $D$ and $D′$ be small catgeories and let $C$ be a category which admits limits of shape $D$ as well as limits of shape $D′$. Then these limits commute with each other, in that
for $F:D^{op} \times D^{' op} →C$ a functor , with corresponding induced functors
$F_D:D^{ ′\ op}→[D^{op},C]$ and $F_{D' } : {D}^{op} \to [D^{'\ op},C]$ then
$\lim F≃\lim_D(\lim_{D′} F_D)≃\lim_{D ′}(\lim_D F_{D′})$.
How do I prove the last two isomorphisms?