Grothendieck construction

Remarks

Example: if $F$ is the constant functor with value a small category $\mathcal{D}$, then $F(f) = \id_{\mathcal{D}}$ for every $f$. A morphism $(c, x) \to (c', x')$ is then a pair $(f, \varphi)$ with $\varphi \colon x \to x'$ in $\mathcal{D}$, so $\int F$ is the product category $\mathcal{C} \times \mathcal{D}$.

Non-example: the disjoint union of the categories $F(c)$, with no morphisms between different fibres, is not $\int F$ when $\mathcal{C}$ has a non-identity morphism $f \colon c \to c'$ and $F(c)$ is non-empty. Clause 2 gives the morphism $(f, \id_{F(f)(x)}) \colon (c, x) \to (c', F(f)(x))$ between two fibres.

When each $F(c)$ is a set seen as a category with only identity morphisms, $\int F$ is the category of elements of $F$. For a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Cat}$ the dual construction takes morphisms $(f, \varphi)$ with $f \colon c \to c'$ and $\varphi \colon x \to F(f)(x')$ in $F(c)$; its projection is a fibration.