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 $(a,b):(A,B,f)\to (A^{\mathrm{\prime }},B^{\mathrm{\prime }},f^{\mathrm{\prime }})$ satisfies

$T(b)\circ f=f^{\mathrm{\prime }}\circ S(a)\mathrm{.}$

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