Notation 9.2.8.5 (04PY)

Notation 9.2.8.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. We let $\widetilde{f} = s^{1}_{1}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ depicted in the diagram

\[ A \xrightarrow {f} B \xrightarrow {\operatorname{id}} B \quad \quad X \xrightarrow {\operatorname{id}} X \xrightarrow {g} Y. \]

Let $\beta : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^3$ denote the morphism of simplicial sets given on vertices by the formulae

\[ \beta (0,0) = 0 \quad \quad \beta (1,0) = 1 = \beta (2,0) \quad \quad \beta (0,1) = 2 = \beta (1,1) \quad \quad \beta (2,1) = 3. \]

Then precomposition with $\beta $ determines a functor $\operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^2 \times \Delta ^1, \operatorname{\mathcal{C}})$, which restricts to a map of Kan complexes $T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$. Concretely, $T$ carries a $3$-simplex

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [ur]^{v} \ar [r]^-{w} & Y } \]

to the morphism in $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{u} & B \ar [d]^{v} \ar [r]^-{\operatorname{id}_{B}} & B \ar [d]^{w} \\ X \ar [r]^-{\operatorname{id}_{X}} & X \ar [r]^-{g} & Y. } \]