Thanks for helping me

yea np

do you follow a lecture or textbook ?

I've been looking at several different algebra books (e.g. Hungerford)

I understand how you found the bijection, but to show that it is natural I need to find a natural transformation to make the square commute, right?

no, the bijection IS the natural transformation

first things first

but don't I separately need to show that it is natural?

wait a sec, let's clarify some things before we march on :D

can you identify the /functors/, between which I constructed the natural transformations?

functors

wait, wasn't the bijection between the sets of morphisms B(F(A),B) and A(A,G(B))?

let's use lowercase letters for objects in A and B

because F(A) doesnt make sense

F : A -> B

so you can only apply F to an object a \in A

ok, then the sets of morphisms B(F(a), b) and A(a, G(b))

for any fixed a, b

yes, but then again not every bijection will do, because we need this /naturality/ thing

yeah, that's what I don't understand

we have two sets, and we have a bijection, but how do we show that it is natural?

the concept we need is /natural transformation/

can you define it?

so the recipe goes as follows

it is a map between functors, and it has two maps, F(a) \arrow G(a') and G(a) \arrow G(a') such that the diagram commutes

not quite

the functors cannot be arbitrary, they need to have the same source and target

yes

so you have say two functors F : A -> B, H : A -> B

and a natural transformation between them assigns to each object a \in A a morphism F(a) -> H(a), which is a morphism in B

not just two

and then we also need the diagrams of the morphisms to commute, correct?

yes, so you get a diagram for each morphism in A

you can try to work out this example:

(later)

the assignment V -> V^** maps a vector space to its double-dual

i'll try that later

but the original problem dealt with sets, which is what confused me

the isomorphism was between sets, I mean

no, we have functors

actually

so thats the next thing

in order to have a natural transformation we need to identify the functors

these are: