What is the concept for a category created by exchanging the roles of objects and morphisms in another category?

Question

Is there a concept for a category $C$ created from another category $B$, by treating the morphisms in $B$ as objects in $C$ (and possibly treating the objects in $B$ as morphisms in $C$)? Does such a concept appear in Categories for the Working Mahematician? I can't find the concept in the book. Thanks.

Answer 1

Let's say we wanted to build a new category $D$ in which $\text{ob}(D) = \text{mor}(C)$. We could start with a discrete category where the only morphisms in $D$ are the identity morphisms. Should there be any other morphisms in $D$?

I don't think there is a canonical way to get any non-indentity morphisms in $D$ from the data of $C$ (unless $C$ is a 2-category or something). Morally, since a morphism includes the data of its source and target, you have already exhausted all of the data of $C$ in building this discrete category $D$, and moreover you have forgotten extra data about $\text{mor}(C)$ including which morphisms were identities and which composed.

Answer 2

Like desiinger points out, there is no obvious way to compose morphisms. There is also no obvious domain and codomain here. Instead, I think the closest thing to what you are talking about is this:

Let $C$ be a category. Let $D$ be the category whose objects are morphisms in $C$. Morphisms $ f \rightarrow g$ in $D$ are pairs of morphisms $(\phi , \psi)$ in $C$ such that $\phi \circ f = g \circ \psi$. This is called the arrow category.