Well, we had last time concluded with the term of the functor.

And in advance, we looked at the composition of functors and identity functors.

That means we looked at categories and functors as a kind of category,

except for the fact that everything is too big, both the collection of objects and the collection of allomorphisms between each object.

In other words, categories are just a kind of category, but for most purposes you can transfer categorical terms without too much consideration to categories.

We'll see that later.

So, we'll see a few examples first.

So, first of all, one that will be presented on the paper.

The quadrion functor, I'll write on set now, it's just as good on any other category with final products.

I'll start now, as long as the expressions for objects are not too complicated, to define the action of the functor in this style.

So I create a whole morphism with domain and codomain.

And when I say what it will be, I have defined the functor on both objects and functions or morphisms.

So, such a thing is created on x cross x or x squared.

So, on the cartesian product of x with itself, y is then of course defined on y cross y.

And here comes f cross f, the function that simply applies to both components f.

So, you have to calculate that this thing is functional.

So, for example, if we now put the identity on x, then of course identity on x cross identity on x comes out.

And that's a nice picture that does not do anything on both components, in other words, that's the identity on x cross x, which I will calculate in elements.

And when I look at the composition, f is composed with g and I square that, then f cross f is of course clipped around.

And that should be the same.

This one here, therefore with functionality, is f cross f composed with g cross g.

I'll put the profiles back.

This is what it looks like while f cross f is running from start to finish.

And you can also calculate this quickly in component form.

Here the left component is followed by first g is applied, then f and here f g is applied.

So, that's really a function, so you can easily calculate these two laws.

Then we saw this list constructor, which is also a function on quantities, for example, this is also more general.

We do not know this one, so it is a picture f from x to y, a picture between the corresponding list data types, list x and list y.

And the picture is exactly this list f that we have already learned, which is our way of writing for the map operation.

And we have calculated on the current list that this is actually a function, that is, that it keeps identity and composition.

So, then...

Now let's take any category C and any object A and I am now making myself

a substitute to construct a function from this object here. And that is...

The application of this function on objects gives me a whole lot.

So that's a function from C to Z. It also has a name.

That's the homfunk top. And what does it do?

When I have a morphism, f from B to C...

Well, where do I want to put that? There will be quantities on this page, and as announced,

this is the morphism quantity from A to B, and here the morphism quantity from A to C.

And something comes here. I write that as C of A, f.

Yes, and that has to be depicted here, a morphism, we say G, and that is depicted on...

I only have one thing that can be written down. I have to write down a morphism from A to C.

Yes, but... No, every programming language and every theorem proof should throw away the definition of warning that a non-declared variable is used on the right.

Exactly. F composes with G. So G goes from A to B, F goes from B to C, so the composition goes from A to C.

Now I ask you to calculate that this is really a function.

Yes? We have to check for the identity, that's the approach.

And the point is C of A, right? Yes, exactly.

That means that the identity for F is here, that is, here is the identity, G is depicted on identity after the name G.

That means that the ID comes somewhere else, not where it is now, but... Okay, we want to write it down there, that's a bit confusing. Then we write it down there, then it is already used on the image, then it is applied to G.

That means that C is the correct amount, that is...