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Example (Kleisli Categories). Let $\operatorname{\mathcal{C}}= [0]$ denote the category having a single object and a single morphism. By virtue of Example, we can identify lax functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ with pairs $(\operatorname{\mathcal{E}}, T)$, where $\operatorname{\mathcal{E}}$ is a category and $T$ is a monad on $\operatorname{\mathcal{E}}$: that is, a functor from $\operatorname{\mathcal{E}}$ to itself which is equipped with unit and multiplication maps

\[ \epsilon : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow T \quad \quad \mu : T \circ T \rightarrow T \]

which endow $T$ with the structure of an associative algebra object of the category of endofunctors $\operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}})$. In this case, the Grothendieck construction $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with the Kleisli category of the monad $T$. Concretely, it can be described as follows:

  • The objects of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ are the objects of the category $\operatorname{\mathcal{E}}$.

  • For every pair objects $X,Y \in \operatorname{\mathcal{E}}$, we have $\operatorname{Hom}_{ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} }( X, Y) = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( X, T(Y) )$.

  • Let $u: X \rightarrow Y$ and $v: Y \rightarrow Z$ be morphisms in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, corresponding to a pair of morphisms $u': X \rightarrow T(Y)$ and $v': Y \rightarrow T(Z)$ in the category $\operatorname{\mathcal{E}}$. Then the composition $v \circ u$ in $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ corresponds to the composite map

    \[ X \xrightarrow {u} T(Y) \xrightarrow { T(v) } (T \circ T)(Z) \xrightarrow { \mu (Z) } T(Z) \]

    in the category $\operatorname{\mathcal{E}}$.