Sorry, I must be confused.  Below is my interpretation of how IsotonicRegression has been written.

If I call fit and then transform, then increasing_ is set on fit, but NOT transform, as desired.

If I call fit_transform, there is no call to fit.  My understanding of the code is that fit's body is duplicated within fit_transform; see:
* L213-L222, L230-233
* L283-L292, L300-305

If we refactor the fit_transform method to behave like other estimators, then we could remove the call to check_increasing.