Comma category, slice category, and coslice category

For functors $S:\mathcal A\to\mathcal C$ and $T:\mathcal B\to\mathcal C$ (Covariant functor, identity functor, composite functor, and contravariant functor), the comma category $(S\downarrow T)$ has objects $(A,B,f)$ with $f:SA\to TB$. A morphism $(a,b):(A,B,f)\to(A',B',f')$ satisfies

$T(b)\circ f=f'\circ S(a).$

Identities and composites are componentwise. Functoriality of $S,T$ shows that the displayed square remains commutative under composition, and the category axioms follow from Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

For an object $C\in\mathcal C$, let $\Delta_C:\mathbf 1\to\mathcal C$ be the functor from the one-object, identity-only category that selects $C$. The slice category is $\mathcal C/C=(1_{\mathcal C}\downarrow\Delta_C)$: its objects are arrows $f:A\to C$, and a morphism from $f:A\to C$ to $f':A'\to C$ is an arrow $a:A\to A'$ with $f'\circ a=f$. The coslice category is $C/\mathcal C=(\Delta_C\downarrow1_{\mathcal C})$: its objects are arrows $f:C\to A$, and a morphism from $f:C\to A$ to $f':C\to A'$ is an arrow $a:A\to A'$ with $a\circ f=f'$.