Why there is a map between $X$ and $Y$ in the definition of limit but not between them in the product definition?

Question

Here is the definition of the product from Tom Leinster book:

And here is the definition of the limit:

My question is:

What is $Du$ in case of the definition of a product? could anyone help me answer this question please?

Answer 1

There is no $Du$ in the case of a product because there is no $u$ (except of course the identity morphisms of $\bf I,$ but for them the triangle trivially commutes since $D$ is a functor).

The category $\bf I$ is just a pair $\{1,2\}$ and no arrow (except ${\rm id}_1$ and ${\rm id}_2$).
That way,

• a diagram $D:{\bf I}\to\mathcal A$ is just given by two objects $X=D(1),Y=D(2)\in\mathcal A,$ and
• a cone on $D$ is an object $A\in\mathcal A,$ together with two morphisms $f_1:A\to X,f_2:A\to Y$ in $\mathcal A,$ with no condition.