Dual of a Comma Category Let $F:\mathcal{C} \longrightarrow \mathcal{D}$ be a functor between two categories and $d$ be an object of  $\mathcal{D}$. Does the following hold:
$(F \downarrow d)^{op} \cong (d \downarrow F^{op})$
If yes, how can I prove this?

I don't think so. The objects of $(F\downarrow d)$ as well as its opposite
are pairs $(c,f)$ where $f:Fc\to d$ is an arrow in $\mathcal{D}$.
The objects of $(d\downarrow F^{op})$ are pairs $(c,g)$ where $g:d\to Fc$
is an arrow in $\mathcal{D}$. There's no natural way to pair off these two different classes of objects.
