1.NounA category built out of a pair of functors that have the same codomain."Given a pair of functors
S
:
A
→
C
{\displaystyle S:{\mathcal {A}}\rightarrow {\mathcal {C}}}
and
T
:
B
→
C
{\displaystyle T:{\mathcal {B}}\rightarrow {\mathcal {C}}}
, objects of the comma category
S
↓
T
{\displaystyle S\downarrow T}
are arrows
h
:
S
(
A
)
→
T
(
B
)
{\displaystyle h:S(A)\rightarrow T(B)}
parametrized by triples (A, B, h), and given morphisms
f
:
A
→
A
′
{\displaystyle f:A\rightarrow A'}
and
g
:
B
→
B
′
{\displaystyle g:B\rightarrow B'}
, then a morphism of the said comma category is a commuting square parametrized by the pair (f, g) and spanning the area from h to
h
′
:
S
(
A
′
)
→
T
(
B
′
)
{\displaystyle h':S(A')\rightarrow T(B')}
and from
S
(
f
)
{\displaystyle S(f)}
to
T
(
g
)
{\displaystyle T(g)}
."