I've stumbled over this question, and saw that it had no answer to the "Implicit Function Theorem" part. Therefore, I wanted to address this in an answer. However, for that, I would need diagonal arrows in a commutative diagram, which seems to be not available in MSE (only through slick means such as drawing diagrams as arrays, which is quite unpleasant imho). Hence, I made the following answer in LaTeX:
and wanted to post it as an answer. This seems to be discouraged by MSE, but it seems to me that another option is unavailable. How to proceed?