linguistic:voice_change, information:filtering
Connection with comma categories

Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right).

Let F: C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor and X {\displaystyle X} an object of D {\displaystyle {\mathcal {D}}}. Then recall that the comma category (X ↓ F) {\displaystyle (X\downarrow F)} is the category where

Now suppose that the object (A, u: X → F (A)) {\displaystyle (A,u:X\to F(A))} in (X ↓ F) {\displaystyle (X\downarrow F)} is initial. Thenfor every object (A ′, f: X → F (A ′)) {\displaystyle (A',f:X\to F(A'))}, there exists a unique morphism h: A → A ′ {\displaystyle h:A\to A'} such that the following diagram commutes.

Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a universal morphism from X {\displaystyle X} to F {\displaystyle F}. Therefore, we see that a universal morphism from X {\displaystyle X} to F {\displaystyle F} is equivalent to an initial object in the comma category (X ↓ F) {\displaystyle (X\downarrow F)}.

Conversely, recall that the comma category (F ↓ X) {\displaystyle (F\downarrow X)} is the category where

Suppose (A, u: F (A) → X) {\displaystyle (A,u:F(A)\to X)} is a terminal object in (F ↓ X) {\displaystyle (F\downarrow X)}. Then for every object (A ′, f: F (A ′) → X) {\displaystyle (A',f:F(A')\to X)}, there exists a unique morphism h: A ′ → A {\displaystyle h:A'\to A} such that the following diagrams commute.

The diagram on the right side of the equality is the same diagram pictured when defining a universal morphism from F {\displaystyle F} to X {\displaystyle X}. Hence, a universal morphism from F {\displaystyle F} to X {\displaystyle X} corresponds with a terminal object in the comma category (F ↓ X) {\displaystyle (F\downarrow X)}.