(Comma category)
Let \(S \colon \mathsf{A} \to \mathsf{C}\) and \(T \colon \mathsf{B} \to \mathsf{C}\) be functors. The comma category \(S \downarrow T\) has as objects the triples \((A, B, h)\) with \(A \in \mathsf{A}\), \(B \in \mathsf{B}\) and \(h \colon S(A) \to T(B)\) a morphism of \(\mathsf{C}\). A morphism \((A_0, B_0, h_0) \to (A_1, B_1, h_1)\) is a pair \((f, g)\) of morphisms \(f \colon A_0 \to A_1\) in \(\mathsf{A}\) and \(g \colon B_0 \to B_1\) in \(\mathsf{B}\) such that \(h_1 \circ S(f) = T(g) \circ h_0\):
Composition is componentwise.
(Slice categories)
Let \(\mathsf{pt}\) be the category with one object \(\ast\) and only its identity morphism, \(X \in \mathsf{A}\), and \(T_X \colon \mathsf{pt} \to \mathsf{A}\) the functor with \(T_X(\ast) = X\). Then \(\id_{\mathsf{A}} \downarrow T_X\) has objects \((A, \ast, h \colon A \to X)\) and morphisms \((f, \id_\ast)\) with \(h_1 \circ f = h_0\), so it is the slice category \(\mathsf{A}_{/X}\) of objects over \(X\).
(Cocones)
Let \(F \colon \mathsf{J} \to \mathsf{C}\) be a functor, and let \(\Delta \colon \mathsf{C} \to \operatorname{Fun}(\mathsf{J}, \mathsf{C})\) be the diagonal functor, sending \(N\) to the constant functor \(\Delta(N)\) with value \(N\) on objects and \(\id_N\) on morphisms. A cocone over \(F\) is an object \(N \in \mathsf{C}\) with a natural transformation \(\psi \colon F \to \Delta(N)\): morphisms \(\psi_X \colon F(X) \to N\) for \(X \in \mathsf{J}\) with \(\psi_Y \circ F(u) = \psi_X\) for every \(u \colon X \to Y\) in \(\mathsf{J}\). The category of cocones over \(F\) is the comma category \(\ast_F \downarrow \Delta\), also written \(F \downarrow \Delta\), where \(\ast_F \colon \mathsf{pt} \to \operatorname{Fun}(\mathsf{J}, \mathsf{C})\) picks out \(F\); its objects are the cocones \((N, \psi)\), and a morphism \((N, \psi) \to (N', \psi')\) is a morphism \(g \colon N \to N'\) with \(g \circ \psi_X = \psi'_X\) for all \(X\).
A colimit of \(F\) (D-5MX7E) is exactly an initial object of the category of cocones over \(F\). In particular a colimit is unique up to unique isomorphism.
For a presheaf \(\mathcal{F}\) on a space \(Y\) and a point \(p \in Y\), the stalk \(\mathcal{F}_p\) (D-0QSI0) is the colimit of the functor \(U \mapsto \mathcal{F}(U)\) on the category of open neighbourhoods of \(p\) with the opposite of inclusion, and the maps \(\mathcal{F}(U) \to \mathcal{F}_p\), \(s \mapsto s_p\), form the initial cocone.