We talked about products in categories and we talked about dualization.

Each term has a dual term and then there are dual theorems.

Now we start to put it together.

We can go and dualize this term of the product.

Now we have time for every five minutes and everyone paints the definition of a co-product on paper.

Then we find a person who writes it on the board.

It's an exercise in turning arrows.

This is an exercise in turning arrows.

This is now.

Most people are not doing anything anymore.

Now we can write it on the board.

This is because it is so ready.

That's why it's so flexible.

This is because it is so ready.

Maybe I should explain the quantification.

This is because it is so ready.

This is because it is so ready.

This is because it is so ready.

We know that the products are clearly defined.

We know that the product is commutative.

We have identity.

We have identity.

We have shown that the initial object is a neutral element for the product.

We know that the initial object is a neutral element for the product.

Let's look at the examples.

This is the first example.

The first example is the first example.

We have to disjunct these quantities.

We have to disjunct these quantities.

We have to write the label.

We have to write the label.

We have to write the label.

We have to write the label.

We have to write the label.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.

We have elements of two forms.