Inherent Risk and Control Risk are independent in the way that you're asking. An increase in IR will not increase CR, nor vice versa. IR x CR is actually the Risk of Material Misstatement (RMM). Now and increase in IR or CR will increase both RMM and Audit Risk (AR).
Inherent risk is how likely a relevant assertion is to be materially misstated in the absence of internal controls.
Control risk is the risk that the Control Activities won't prevent or detect those misstatements.
Put another way, IR is the likelihood that a misstatement will occur, while CR is the likelihood that Internal Controls won't prevent or detect the misstatement when it does occur.
An increase in Inherent Risk would not affect Control Risk, but it is part of the consideration in determining the N,T,E of the audit procedures to obtain evidence that controls are working properly.
Remember, the auditor can only change Detection Risk in response to IR and CR. This is done through Substantive Procedures. Higher IR x CR (RMM), the more reliable, relevant evidence you need to reduce your Detection Risk. One trick to remember this is to rearrange the audit risk formula to isolate DR, like this:
AR/IRxCR = DR
This also helps to demonstrate the inverse relationship DR has to both IR and CR.
IR and CR are given, they exist as part of the client's business. AR is a judgment call by the auditing firm. It's how much risk they're willing to take that they will sign off on materially misstated financial statements. If the nature and environment of the client's business is risky, and their Internal Control is weak, the only thing the auditor can do in response is modify the Nature, Timing, and Extent of their audit procedures to reduce both DR and AR.
Hope this helps!
.