Leinster question on isomorphic functor categories Hi so the question is basically to prove that for categories A and B
$$
[A,B]^{op} \simeq [A^{op},B^{op}]
$$
where these are functor categories.
So my first port of call was to write out what the structure of these beasts are and then to try and conjure up the relevant isomorphisms. Alas here I am.
First up I noted that 
$[A,B]^{op}$ has the same objects as $[A,B]$ , namely functors $F,G:A \rightarrow B$, but here the morphisms would be reversed i.e the natural transformations $\alpha:F\rightarrow G$ in $[A,B]$ become $\tilde{\alpha}:G\rightarrow F$ in $[A,B]^{op}$
Secondly for $[A^{op},B^{op}]$ I thought that for this category objects are functors $H,I:A^{op}\rightarrow B^{op}$ and morphisms are some natural transformations between them $\alpha':A^{op} \rightarrow B^{op}$ merrily mapping the inverse morphisms of $A$ and $B$ to each other.
So I feel like there is some deeper work to be done on this second point on relating it to $[A,B]$ in a concrete and explicit manner so an isomorphism could be explicitly constructed. This is the end of the road as far as my thought path goes however. I'm aware that these problems usually solve themselves on having a clear understanding of exactly what is entailed in the question; any help to achieving this would be greatly appreciated. Of course for bonus points the rigorous proof is the dream response.

Look at what a morphism $\alpha:F\to G$ looks like in $[A^o,B^o]$. It is a collection $\{\alpha_X:FX\to GX \}$ of morphisms in $B^o$, i.e., a collection $\{\bar{\alpha}_X:GX \to FX\}$ of morphisms in $B$.
Therefore, if you want to identify $[A^o,B^o]$ and $[A,B]^o$, a good idea would be to send $\alpha$ to $\bar{\alpha}$. 
Now, you should verify that $\alpha$ is a natural transformation if and only if $\bar{\alpha}$ is.
