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Remark In the situation of Example, the Kleisli category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ is equipped with a comparison functor $H: \operatorname{\mathcal{E}}\rightarrow \int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, which carries each object of $\operatorname{\mathcal{E}}$ to itself and each morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{E}}$ to the composite morphism $X \xrightarrow {u} Y \xrightarrow { \epsilon (Y) } T(Y)$ (regarded as a morphism from $X$ to $Y$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$). The following conditions are equivalent:

  • The functor $H$ is an equivalence of categories.

  • The functor $H$ is an isomorphism of categories.

  • The unit map $\epsilon : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow T$ is an isomorphism of functors.