Let $\mathcal A$ and $\mathcal B$ be categories. I want to prove that $[\mathcal A^\mathrm{op},\mathcal B^\mathrm{op}]$ is isomorphic with $[\mathcal A,\mathcal B]^\mathrm{op}$.
Thus, I want to find two functors $i:[\mathcal A^\mathrm{op},\mathcal B^\mathrm{op}]\to[\mathcal A,\mathcal B]^\mathrm{op}$ and $j:[\mathcal A,\mathcal B]^\mathrm{op}\to[\mathcal A^\mathrm{op},\mathcal B^\mathrm{op}]$ such that $i\circ j$ and $j\circ i$ are identity functors.
A think that $i$ can be defined in the following way:
If $F:\mathcal A^\mathrm{op}\to\mathcal B^\mathrm{op}$ is functor, then $i(F)(f)=(F(f^\mathrm{op}))^\mathrm{op}$ for every morphism $f$ in $\mathcal A$.
If $\alpha=(\alpha_A)_{A\in \mathcal A^\mathrm{op}}$ is morphism in $[\mathcal A^\mathrm{op},\mathcal B^{\mathrm{op}}]$, i.e. natural transformation, then $i(\alpha)=(\alpha_A^{\mathrm{op}})_{A\in \mathcal A}$.
And I also think that $j$ can be defined in similar ("dual") manner.
Am I correct?