The Stacks project

Definition 7.8.2. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$, and $\mathcal{V} = \{ \psi _ j : V_ j \to U\} _{j\in J}$ be two families of morphisms with fixed target.

1. We say $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ such that $\varphi _ i = \psi _{\alpha (i)}$ and $\psi _ j = \varphi _{\beta (j)}$.

2. We say $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ and for all $i\in I$ and $j \in J$ commutative diagrams

   \[ \xymatrix{ U_ i \ar[rd] \ar[rr] & & V_{\alpha (i)} \ar[ld] & & V_ j \ar[rd] \ar[rr] & & U_{\beta (j)} \ar[ld] \\ & U & & & & U & } \]

   with isomorphisms as horizontal arrows.