Kerodon

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Remark 2.3.6.5. Let $(Q, \leq )$ be a partially ordered set, let $T_{Q}: Q \rightarrow \operatorname{Path}_{(2)}[Q]$ be the lax functor of Construction 2.3.6.4, and let $F: \operatorname{Path}_{(2)}[Q] \rightarrow Q$ be the functor of Remark 2.3.6.3 (so that $F$ is the identity on objects). Then the composition

\[ Q \xrightarrow {T_ Q} \operatorname{Path}_{(2)}[Q] \xrightarrow {F} Q \]

is the identity functor from $Q$ to itself. Beware that the composition

\[ \operatorname{Path}_{(2)}[Q] \xrightarrow {F} Q \xrightarrow {T_ Q}\operatorname{Path}_{(2)}[Q] \]

is not the identity (as a lax functor from $\operatorname{Path}_{(2)}[Q]$ to itself). This composition carries each object of $\operatorname{Path}_{(2)}[Q] $ to itself, but is given on $1$-morphism by the construction $\{ x_0 < x_1 < \cdots < x_ n \} \mapsto \{ x_0 < x_ n \} $.