Suppose S and T are functors with the same codomain. Then you can construct the *comma category* . Let C be the common codomain of S and T, let D be the domain of S, and let E be the domain of T. Suppose X in D, Y in E, and f: S(X) -> T(Y) in C. Then (X,f,Y) is an object of . Suppose (Z,g,W) is another object of . Suppose h: X -> Z in D and i: Y -> W in E. Suppose T(i).f = g.S(h), for a commuting square in C. Then (f,h,i,g): (X,f,Y) -> (Z,g,W) is a morphism of . Note that the domain of (f,h,i,g) is (dom h, f, dom i) and the codomain of (f,h,i,g) is (codom h, g, codom i). Suppose (g,k,l,j): (Z,g,W) -> (V,j,U) is another morphism of . Then the composition (g,k,l,j).(f,h,i,g) is (f,k.h,l.i,j). Suppose D is the empty category 0. Since 0 has no objects, has no objects. In other words, <0 -> C, T> = 0. Suppose E is the empty category 0. Since 0 has no objects, has no objects. In other words, C> = 0. Suppose C is the empty category 0. Then S and T may be identified with 0. Since C has no morphisms, <0,0> has no objects. In other words, <0,0> = 0. We already knew this, of course, since D and E would have to be 0. Suppose D is a singleton, a category with a unique object and morphism. Then S may be identified with an object X of C. Suppose Y in E and f: X -> T(Y) in C. Then (f,Y) is essentially an object of . Suppose (g,W) is another object of . Suppose i: Y -> W in E. Suppose T(i).f = g. Then (f,i,g): (f,Y) -> (g,W) is a morphism of . Note that the domain of (f,i,g) is (f, dom i) and the codomain of (f,i,g) is (g, codom i). Suppose (g,l,j): (g,W) -> (j,U) is another morphism of . Then (g,l,j).(f,i,g) is (f,l.i,j). This is the usual kind of comma category. Suppose E is a singleton, so T is identified with an object Y of C. Suppose X in D and f: S(X) -> Y in C. Then (X,f) is an object of . Suppose (Z,g) is another object of . Suppose h: X -> Z in D. Suppose f = g.S(h). Then (f,h,g): (X,f) -> (Z,g) is a morphism of . Note that the domain of (f,h,g) is (dom h, f) and the codomain of (f,h,g) is (codom h, g). Suppose (g,k,j): (Z,g) -> (V,j) is another morphism of . Then (g,k,j).(f,h,g) is (f,k.h,j). This is the usual kind of cocomma category. Suppose D and E are both singletons, so S and T are identified with objects X and Y of C. Suppose f: X -> Y in C. Then f is an object of . Suppose g is another object of . The only morphism f -> g in is (f,g), and that exists only if f = g. In other words, is the set of morphisms from X to Y. Suppose C is a singleton. Suppose X in D and Y in E. Then (X,Y) is an object of . Suppose (Z,W) is another object of . Suppose h: X -> Z in D and i: Y -> W in E. Then (h,i): (X,Y) -> (Z,W) is a morphism of . Note that the domain of (h,i) is (dom h, dom i) and the codomain of (h,i) is (codom h, codom i). Suppose (k,l): (Z,W) -> (V,U) is another morphism of . Then (k,l).(h,i) is (k.h,l.i). In other words, 1, E -> 1> is the Cartesian product of D and E. Suppose S is the identity functor on C. Then S may be identified with C. Suppose X in C, Y in E, and f: X -> T(Y) in C. Then (f,Y) is an object of . Suppose (g,W) is another object of ; let Z be dom g. Suppose h: X -> Z in C and i: Y -> W in E. Suppose T(i).f = g.h. Then (f,h,i,g): (f,Y) -> (g,W) is a morphism of . Note that the domain of (f,h,i,g) is (f, dom i) and the codomain of (f,h,i,g) = (g, codom i). I suppose could be called the slice category of C over T. Suppose T is the identity functor on C, so T is identified with C. Suppose X in D, Y in C, and f: S(X) -> Y in C. Then (X,f) is an object of . Suppose (Z,g) is another object of ; let W be codom g. Suppose h: X -> Z in D and i: Y -> W in C. Suppose i.f = g.S(h). Then (f,h,i,g): (X,f) -> (Z,g) is a morphism of . Note that the domain of (f,h,i,g) is (dom h, f) and the codomain of (f,h,i,g) is (codom h, g). I suppose could be called the coslice category of C under S. Suppose S and T are both the identity functor on C, so they are both identified with C. Suppose X and Y in C and f: X -> Y in C. Then f is an object of . Suppose g is another object of ; let Z be dom g, and let W be codom g. Suppose h: X -> Z and i: Y -> W in C. Suppose i.f = g.h. Then (f,h,i,g): f -> g is a morphism of . Note that the domain of (f,h,i,g) is f and the codomain of (f,h,i,g) is g. In other words, is the arrow category of C. Suppose S is the identity functor on C and E is a singleton, so S is identified with C and T is identified with an object Y in C. Suppose X in C and f: X -> Y in C. Then f is an object of . Suppose g is another object of ; let Z be dom g. Suppose h: X -> Z is a morphism of C. Suppose f = g.h. Then (f,h,g): f -> g is a morphism of . Note that the domain of (f,h,g) is f and the codomain of (f,h,g) is g. Suppose (g,k,j): g -> j is another morphism of . Then (g,k,j).(f,h,g) is (f,k.h,j). In other words, is the slice category of C over Y. Suppose D is a singleton and T is the identity functor on C, so S is identified with an object X in C and T is identified with C. Suppose Y is an object of C and f: X -> Y is a morphism of C. Then f is an object of . Suppose g is another object of ; let W be codom g. Suppose i: Y -> W is a morphism of C. Suppose i.f = g. Then (f,i,g): f -> g is a morphism of . Note that the domain of (f,i,g) is f and the codomain (f,i,g) is g. Suppose (g,l,j): g -> j is another morphism of . Then (g,l,j).(f,i,g) is (f,l.i,j). In other words, is the coslice category of C under X. Suppose C is a singleton and D = C. Then S may be identified with 1. Suppose Y is an object of E. Then Y is an object of <1,T>. Suppose W is another object of <1,T>. Suppose i: Y -> W in E. Then i: Y -> W is a morphism of <1,T>. Note that the domain and codomain of i in <1,T> are the same as in E. Suppose l: W -> U is another morphism of <1,T>. Then l.i is the same in <1,T> as in E. In other words, <1, E -> 1> is the category E. We already knew this, since 1 x E = E. Suppose C is a singleton and C = E, so T is identified with 1. Suppose X is an object of D. Then X is an object of . Suppose Z is another object of . Suppose h: X -> Z in D. Then h: X -> Z is a morphism of . Note that the domain and codomain of h in are the same as in D. Suppose k: Z -> V is another morphism of . Then k.h is the same in as in D. In other words, 1, 1> is the category D. We already knew this, since D x 1 = D. Suppose C is a singleton and D = C = E, so S and T are identified with 1. There is a unique object of <1,1>. There is a unique morphism of <1,1>. In other words, <1,1> is a singleton. We already knew this, of course, in several ways.