Definition 4.44.1. Let \mathcal{C} be a (2,1)-category. Consider a 2-commutative solid diagram
4.44.1.1
\begin{equation} \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 } } \end{equation}
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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