Construction 8.1.3.12. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts, and let $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ denote the $2$-category of Example 2.2.2.1. Suppose we are given another category $\operatorname{\mathcal{D}}$ and a functor $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. We define a strictly unitary lax functor $F^{+}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ as follows:
For each $X \in \operatorname{\mathcal{D}}$, we define $F^{+}(X) = F( \operatorname{id}_{X} )$; here we regard the identity morphism $\operatorname{id}_ X: X \rightarrow X$ as an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$.
For each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{D}}$, we define $F^{+}(f)$ to be the $1$-morphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ given by the cospan
\[ F( \operatorname{id}_ X ) \xrightarrow { F(\operatorname{id}_ X, f) } F(f) \xleftarrow { F( f, \operatorname{id}_ Y) } F( \operatorname{id}_ Y ). \]Note that this determines the values of $F^{+}$ on $2$-morphisms, since every $2$-morphism in $\operatorname{\mathcal{D}}$ is an identity $2$-morphism.
For every pair of composable morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{D}}$, the composition constraint $\mu _{g,f}: F^{+}(g) \circ F^{+}(f) \Rightarrow F^{+}(g \circ f)$ is the $2$-morphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ corresponding to the map $F(f) \amalg _{ F( \operatorname{id}_ Y) } F(g) \rightarrow F( g \circ f )$ classifying the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ F( \operatorname{id}_ Y ) \ar [r]^-{ F(f,\operatorname{id}_ Y)} \ar [d]^{F(\operatorname{id}_ Y,g)} & F( f ) \ar [d]^{F(\operatorname{id}_ X,g)} \\ F( g ) \ar [r]^-{F(f,\operatorname{id}_ Z)} & F( g \circ f) } \]in the category $\operatorname{\mathcal{C}}$.