Well, let's try to continue with insights that enable the solution of task 4 and task 5.

So, we are dealing with natural transformations.

So, we are now moving one level further.

We have found that we can reflect the category theory on the basis of size problems.

So, the categories form a category again by using the functions as morphisms between categories.

Functions themselves form a category again by using natural transformations as morphisms between functions.

This is a point where you get a headache, especially when you want to paint.

For example, I have two categories, C and D.

And then I have two functions between these two categories, let's say F and G.

And a natural transformation from F to G is a morphism in this function category from C to D.

We'll say what that is, but that's how you paint it.

So, I have two levels of morphisms here.

In practice, this is not what we do, but in extreme cases, there are three levels.

So, the categories where this all takes place, if D is built in such a way that morphisms from D exist again,

then there is a third level, which is also a morphism between natural transformations, which are called modifications.

We don't want to do that here, so it stays at two levels, so categories, functions, natural transformations.

Alternatively, we can simply write alpha from F to G, and if we want to make it clearer,

in which function category this takes place, we would again give the profile to G, so G from C to D.

Natural transformations are always only between functions of the same profile,

so they are always only between functions that have the same output and target category.

So, and now comes the actual definition.

So, a natural transformation is a family of morphisms alpha C in D,

from Fc to Gc, and the whole thing is indicated by the objects of C.

So, for each object of C, I get one morphism in D from Fc to Gc.

I'll leave the indication aside, I'll write it instead.

So, indicated by objects of C, I can also write that alternatively to the family.

So, with, and now comes the diagram that they have to fulfill.

So, I have one morphism here, and let's say I have a C morphism, F from C to D,

which is a demorphism Ff from Fc to Fd, but I can also write G on it,

then it becomes Gf from Gc to Gd, and down here then runs alpha D. So, and this thing should commutate.

And this equation here, is called naturalness.

So, that can be done like this, it will become clearer in examples what it is and what it should be.

I will first complete the definition, I have now said what morphisms of functions are.

A self-respecting morphism should be composable with other morphisms,

and there should be identity morphisms, so that the whole thing results in a category again.

That means, I hold on, I have an identity transformation, itf from F to F,

with of course, itf index C, I have to say, what is this thing on index C,

that is like itfc, where I have only left the brackets here, that means, here it is simply applied to Fc,

that is the identity on Fc, and if I say, as before, I have alpha from F to G,

and another transformation beta from G to H, then I get,

a natural transformation from F to H, of course, I write beta composed with alpha,

in the usual way, and it also has components at each C,

and I get them by transformation of the corresponding components, so here alpha C, here beta C,

so that means, I get here beta alpha from F to H, so in other words, I have a composition,

I have to calculate, of course, that this so-defined beta alpha will fulfill the naturalness again,

that it will allow the naturalness diagram to be commuted.

So, let's go there, I have my morphism F again,

and I have Ff, so Ff from Fc to Fd, and here of course, Hf from Hc to Hd,

so, and I have to show, if I put the beta alpha D down here,

then this outer square commutes here.