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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