IIRC, their dual mandate is:

"Price stability": sounds like no inflation or deflation to me, but in fact they pursue some non-zero amount of inflation, usually 2-3%.

"Full employment": sounds like 0% unemployment to me, but in fact they target some non-zero value, around 5% I think.

Targeting equity prices or other asset prices is not officially within their mandates, but given their behavior at times (e.g. suddenly dropping dovish comments when the market drops), one might start to think this is a de facto or shadow mandate. :)

I think worrying about the price of the dollar is closely related to the level of inflation or uh "price stability" they are aiming for.