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Construction 8.1.5.2. Let $\operatorname{\mathcal{A}}$ be a category, let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$, and let $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ be a functor. We define a strictly unitary lax functor $U_ F: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$ as follows:

$(1)$

The lax functor $U_{F}$ is given on objects by $U_ F(0,A) = X$ and $U_ F(1,A) = Y$ for each object $A \in \operatorname{\mathcal{A}}$.

$(2)$

Let $f: A \rightarrow B$ be a morphism in the category $\operatorname{\mathcal{A}}$, which we also regard as an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{A}})$. For $0 \leq i \leq j \leq 1$, we let $f_{ji}$ denote the corresponding morphism from $(i,A)$ to $(j,B)$ in the product category $[1] \times \operatorname{\mathcal{A}}$. Then the lax functor $U_{F}$ is given on $1$-morphisms by the formula

\[ U_{F}( f_{ji} ) = \begin{cases} \operatorname{id}_{X} & \text{ if } i = j = 0 \\ \operatorname{id}_{Y} & \text{ if } i = j = 1 \\ F(f) & \text{ if } 0 = i < j = 1. \end{cases} \]
$(3)$

Let $f: A \rightarrow B$ and $v: B \rightarrow C$ be composable morphisms in the category $\operatorname{\mathcal{A}}$, and let $0 \leq i \leq j \leq k \leq 1$. Then the composition constraint $\mu _{g_{kj}, f_{ji} }$ for the lax functor $U_ F$ is given as follows:

  • If $i=j=k=0$, then $\mu _{g_{kj},f_{ji} }$ is the unit constraint $\upsilon _{X}: \operatorname{id}_{X} \circ \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}_{X}$ of the $2$-category $\operatorname{\mathcal{C}}$.

  • If $i=0$ and $j=k=1$, then $\mu _{g_{kj},f_{ji} }$ is given by the composition

    \[ \operatorname{id}_{Y} \circ F(f) \xRightarrow { \lambda _{F(f)} } F(f) \xRightarrow { F(\operatorname{id}_ A, g) } F( g \circ f), \]

    where $\lambda _{F(f)}$ is the left unit constraint of Construction 2.2.1.11 and we regard the pair $(\operatorname{id}_ A,g)$ as an element of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}( f, g \circ f)$.

  • If $i=j=0$ and $k=1$, then $\mu _{g_{kj},f_{ji} }$ is given by the composition

    \[ F(g) \circ \operatorname{id}_{X} \xRightarrow { \rho _{F(g)} } F(g) \xRightarrow { F(f, \operatorname{id}_ C) } F( g \circ f), \]

    where $\rho _{F(g)}$ is the right unit constraint of Construction 2.2.1.11 and we regard the pair $(f, \operatorname{id}_ C)$ as an element of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}(g, g \circ f)$.

  • If $i=j=k=1$, then $\mu _{g_{kj},f_{ji} }$ is equal to the unit constraint $\upsilon _{Y}: \operatorname{id}_{Y} \circ \operatorname{id}_{Y} \xRightarrow {\sim } \operatorname{id}_{Y}$ of the $2$-category $\operatorname{\mathcal{C}}$.