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In the example, $I$ is the inclusion functor from the category of integers $\mathbb{Z}$ to the category of real numbers $\mathbb{R}$, and $F$ is the floor-function functor from $\mathbb{R}$ to $\mathbb{Z}$. The lecturer states that $I$ is left adjoint to $F$.

Conditions: Defined between the two order-categories $\mathbb{Z}$ and $\mathbb{R}$.; $I$ is the inclusion functor $\mathbb{Z}\to\mathbb{R}$.; $F$ is the floor-function functor $\mathbb{R}\to\mathbb{Z}$.

In the example, $I$ is the inclusion functor from the category of integers $\mathbb{Z}$ to the category of real numbers $\mathbb{R}$, and $F$ is the floor-function functor from $\mathbb{R}$ to $\mathbb{Z}$. The lecturer states that $I$ is left adjoint to $F$.

Conditions: Defined between the two order-categories $\mathbb{Z}$ and $\mathbb{R}$.; $I$ is the inclusion functor $\mathbb{Z}\to\mathbb{R}$.; $F$ is the floor-function functor $\mathbb{R}\to\mathbb{Z}$.

The category $\mathbb{R}$ is defined as a poset category where the objects are real numbers, and there is a morphism $x \to y$ exactly when $x \le y$. This means the morphisms encode the ordinary order relation on the real numbers.

Conditions: Objects are real numbers.; A morphism $x\to y$ exists precisely when $x\le y$.

The category $\mathbb{R}$ is defined as a poset category where the objects are real numbers, and there is a morphism $x \to y$ exactly when $x \le y$. This means the morphisms encode the ordinary order relation on the real numbers.

Conditions: Objects are real numbers.; A morphism $x\to y$ exists precisely when $x\le y$.