Corollary 3

Let $A, B, C$ be three rings, $S$ (resp. $T, U$) a multiplicative subset of $A$ (resp. $B, C$), $f : A \to B$, $g : B \to C$ two homomorphisms and $h : A \to C$ the composite homomorphism $g \circ f$; suppose that $f(S) \subset T$, $g(T) \subset U$. Let $f' : A[S^{-1}] \to B[T^{-1}]$, $g' : B[T^{-1}] \to C[U^{-1}]$, $h' : A[S^{-1}] \to C[U^{-1}]$ the homomorphisms corresponding to $f, g, h$; then $h' = g' \circ f'$.

This follows easily from the definitions.

In particular, if $S, T, U$ are three multiplicative subsets of $A$ such that $S \subset T \subset U$, then $i_A^{U,S} = i_A^{U,T} \circ i_A^{T,S}$.