comma category

Given a pair of functors S:𝒜→𝒞 and T:ℬ→𝒞, objects of the comma category S↓T are arrows h:S(A)→T(B) parametrized by triples (A, B, h), and given morphisms f:A→A' and g:B→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') and from S(f) to T(g).