Definition 7.8.1. Let \mathcal{C} be a category. Let \mathcal{U} = \{ U_ i \to U\} _{i\in I} be a family of morphisms of \mathcal{C} with fixed target. Let \mathcal{V} = \{ V_ j \to V\} _{j\in J} be another.
A morphism of families of maps with fixed target of \mathcal{C} from \mathcal{U} to \mathcal{V}, or simply a morphism from \mathcal{U} to \mathcal{V} is given by a morphism U \to V, a map of sets \alpha : I \to J and for each i\in I a morphism U_ i \to V_{\alpha (i)} such that the diagram
\xymatrix{ U_ i \ar[r] \ar[d] & V_{\alpha (i)} \ar[d] \\ U \ar[r] & V }is commutative.
In the special case that U = V and U \to V is the identity we call \mathcal{U} a refinement of the family \mathcal{V}.
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