Definition 4.44.1. Let $\mathcal{C}$ be a $(2,1)$-category. Consider a $2$-commutative solid diagram

4.44.1.1
$$\label{categories-equation-dotted-arrows} \vcenter { \xymatrix{ S \ar[r]_-x \ar[d]_ j & X \ar[d]^ f \\ T \ar[r]^-y \ar@{..>}[ru] & Y } }$$

in $\mathcal{C}$. Fix a $2$-isomorphism

$\gamma : y \circ j \rightarrow f \circ x$

witnessing the $2$-commutativity of the diagram. Given (4.44.1.1) and $\gamma$, a dotted arrow is a triple $(a, \alpha , \beta )$ consisting of a morphism $a \colon T \to X$ and and $2$-isomorphisms $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$ such that $\gamma = (\text{id}_ f \star \alpha ) \circ (\beta \star \text{id}_ j)$, in other words such that

$\xymatrix{ & f \circ a \circ j \ar[rd]^{\text{id}_ f \star \alpha } \\ y \circ j \ar[ru]^{\beta \star \text{id}_ j} \ar[rr]^\gamma & & f \circ x }$

is commutative. A morphism of dotted arrows $(a, \alpha , \beta ) \to (a', \alpha ', \beta ')$ is a $2$-arrow $\theta : a \to a'$ such that $\alpha = \alpha ' \circ (\theta \star \text{id}_ j)$ and $\beta ' = (\text{id}_ f \star \theta ) \circ \beta$.

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