Comparison composites derived transport

Observation 20.1.11. The comparison map in part (2) of Definition 20.1.10 is obtained from absoluteness. Write \[ \epsilon _f\colon \bL f_!\circ \gamma _s\Rightarrow \gamma _t\circ f_!, \qquad \epsilon _g\colon \bL g_!\circ \gamma _t\Rightarrow \gamma _u\circ g_! \] for the structure transformations. Since \(\bL f_!\) is absolute, the composite \(\bL g_!\circ \bL f_!\) is a right Kan extension of \(\bL g_!\circ \gamma _t\circ f_!\) along \(\gamma _s\). The composite transformation \[ \bL g_!\circ \bL f_!\circ \gamma _s \Rightarrow \bL g_!\circ \gamma _t\circ f_! \Rightarrow \gamma _u\circ g_!\circ f_! \] therefore induces, by the universal property of \(\bL (g\circ f)_!\), the canonical comparison \[ \bL g_!\circ \bL f_!\longrightarrow \bL (g\circ f)_!. \]