Proposition 5.6.1.15. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the forgetful functor. Then a morphism $(f,u): (C,X) \rightarrow (D,Y)$ of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is $U$-cocartesian if and only if $u: \mathscr {F}(f)(X) \rightarrow Y$ is an isomorphism in the category $\mathscr {F}(D)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume first that $u$ is an isomorphism; we wish to show that $(f,u)$ is a $U$-cocartesian morphism of the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Fix a morphism $g: D \rightarrow E$ of $\operatorname{\mathcal{C}}$ and an object $Z \in \mathscr {F}(E)$; we wish to show that every morphism $(g \circ f,w): (C,X) \rightarrow (E,Z)$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be written uniquely as a composition $(g,v) \circ (f,u)$ for some morphism $(g,v): (D,Y) \rightarrow (E,Z)$. Unwinding the definitions, we wish to show that there is a unique morphism $v: \mathscr {F}(g)(Y) \rightarrow Z$ in the category $\mathscr {F}(E)$ for which the composition
\[ \mathscr {F}( g \circ f)(X) \xrightarrow { \mu _{g,f}^{-1}(X) } (\mathscr {F}(g) \circ \mathscr {F}(f))(X) \xrightarrow { \mathscr {F}(g)(u) } \mathscr {F}(g)(Y) \xrightarrow {v} Z \]
is equal to $w$. This is clear, since $\mu _{g,f}^{-1}(X)$ and $\mathscr {F}(g)(u)$ are isomorphisms.
Now suppose that $(f,u)$ is a $U$-cocartesian morphism of the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$; we wish to show that $u$ is an isomorphism. Let $\iota : \mathscr {F}(D) \rightarrow \{ D\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be the isomorphism of Remark 5.6.1.12. Then the morphism $(f,u)$ factors as a composition
\[ (C,X) \xrightarrow { (f,\operatorname{id}) } (D, \mathscr {F}(f)(X)) \xrightarrow { \iota (u) } (D,Y). \]
The first half of the argument shows that the morphism $(f, \operatorname{id})$ is also $U$-cocartesian, so that $\iota (u)$ is an isomorphism in the fiber $\{ D\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Since $\iota $ is an isomorphism of categories, it follows that $u$ is an isomorphism in the category $\mathscr {F}(D)$. $\square$