I’m sorry if this has already been answered before but one thing that has been bothering me in this series (from the many) is Sherlock’s immediate acceptance of Mary in the TEH itself. The Sherlock Im familiar with would not warm up to a person so quickly especially after deducing she’s a liar. Moffat explained this as him being blind to it because he wanted to like her. But why? Because John likes her? Instead I would’ve expected him to be wary of her because she’s so close to John.

Another good question!

I think it’s beyond doubt that Sherlock wanted to like and be liked by Mary for John’s sake.  And this wasn’t necessarily an honest personal like as much as a ‘professional working relationship’ sort of like.  If John needed her, then by the property of inheritance, Sherlock needed her.

But why would Sherlock care even that much, when in the past he seems to have delighted in driving people off and keeping John’s affections for himself?

Well…we don’t know.  I can see many possibilities.

1: Maybe Sherlock was never opposed in principle to John finding the right woman—but he could tell that none of the women he’d been dating would be right for John.  (Note: what Sherlock considers ‘right’ for John may be…idiosyncratic.)

2: Maybe Sherlock was feeling guilty enough over what he’d done to John that he simply tried to accept Mary as part of an overall apologetic ‘make John happy’ package.

3: Maybe he recognized that, since he interrupted John’s attempt to propose, it was already too late for him to do anything but suck it up and accept John’s choice.

4: Maybe, even though Sherlock was consciously repressing his tendency to deduce people, much like John he intuitively sensed Mary’s profound weirdness and gravitated toward it.

There might even be others.  But the bottom line is, does it make rational sense for Sherlock to blow off all his caution and rational approaches when it comes to Mary?  No.  Is Sherlock in fact a human being whose emotions, fears, doubts and joys sometimes overwhelm his rational faculties despite his best attempts and what he likes to believe?  Yes.  Was he, at the time he met Mary, profoundly emotionally compromised and hella distracted by the danger of losing his best friend?  You betcha.

In short:  I find the likelihood that Sherlock was ever thinking clearly this season when it came to John or anything connected to John to be very low, and when people are emotionally muddled, they make stupid mistakes.  Even Sherlock.

It’s why he normally tries to stick to logic.  Caring is a disadvantage.