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

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

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

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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