ScalingStacks

Suppose we are given a commuting square of spaces where the top horizontal map is an equivalence and where for every point \(c \in C\), the induced map of fibers \(\mathrm{fib}_c(f) \rightarrow\mathrm{fib}_{h(c)}(g)\) is an equivalence. Then, the induced map \[\mathrm{Im}(f) \rightarrow\mathrm{Im}(g)\] is an equivalence.