Sorry this turned out very long, hopefully then it's not a drip feed.
First off, I don't believe she has any idea that she was the OW.
My partner and I split up in March of this year. We had been together for four years but in the last six months his behaviour was so hot and cold with me, which he blamed on the stress of his ongoing divorce, that I had to end it. I'm in France, the divorce laws are different and he's been trying to get a divorce for 5 years, but his ex won't agree to it.
By the time I ended it he had gaslit me so much I thought I was going mad, and had asked the doctor for anti anxiety medication. I honestly thought he was seeing someone else, but whenever I asked he said no, and then we had great days together until he repeated the behaviour. I have never been allowed on his social media, and he wasn't on mine either. It's only with the benefit of hindsight that I realise I was being gas lit. I honestly thought my own insecurities were driving me crazy.
At the beginning of March he went on a "work trip" for the weekend but I couldn't get over the feeling that he was with someone else. I read the relationship boards so I know once the trust is gone end the relationship. So I did when he came back.
Since then I can honestly say I've had an amazing year and I've been very happy, no anxiety, no sleepless night, no migraines. However, every 10/15 days he has sent me a message. It could be how much he misses me, it could be how he still loves me or it could just be something chatty. I have responded but never pursued the conversation.
In June he asked to meet and actually I really wanted to know why he had treated me badly, but we just skirted around the issue. We had a nice drink and chat together, we have always got on well together and I do miss my friend. Afterwards, he sent me a message saying how beautiful I'd looked and how great it had been. I've now seen on his Instagram it was 3 days after he had had a really nice date on a boat with his GF.
At the beginning of September I was on Instagram and he came up as someone I might be interested in. Of course I snooped I'd never been allowed near his Facebook. There wasn't much on there, mainly all pictures of scenery. But I noticed one from December last year of a shopping trip in a nearby city. We didn't do that together. There was a comment on there from a woman saying what a great day they'd had. So I got snooping on his Insta and her FB and actually cross referenced a number of dates together. I did send him a message and asked if he had been seeing anyone at the same time as me, and he said no. I've been busy with work until this week, so on a lonely evening I cross checked his instagram, her FB and the texts he sent me saying why we couldn't be together that day, I did a real deep dive, and all the time I was doing this I was telling myself I was going to be hurt. The place that he had gone to for the work weekend just happens to be the town she originates from and her family are. Could be a coincidence, but I don't think so.
I bumped into him yesterday in town and he asked me if I wanted a coffee. I asked him to his face if he had begun seeing someone when we were still together and if he was seeing her now. I wasn't angry but because he'd gaslit me so much and I thought I was going crazy, I still need to know that it wasn't in my imagination. It was a pleasant conversation, but he said no, and maintained it was all the stress of his divorce. He said again, that he still loved me, regretted that we'd split up but understood my feelings. Even though I'd seen information with my own eyes, I started to believe that I'd got it wrong again. (Because I'm an idiot and he is a very, very good liar).
Then his telephone rang, I saw it was her surname on the screen, he ignored the call and just continued talking. Instead of taking it up with him then when I should have done, I said goodbye and left. As we left, he told me he would still send me messages, because he hoped one day we would find our way back together.
After a glass of wine last night, I sent him a message and asked who was C? That I'd seen they had shared dates while he and I were still together and why had he not been honest when I asked. He replied with an angry face emoticon 
Anyhow, on discussing with my friend she thinks I should send this woman a message and explain how we had been a couple when she got together with him and how even though we were no longer together he still messages me, not always declaring love, but marking his presence in my life. My friend asked me if the shoe was on the other foot would I want to know, and I know that I would. I think this woman has every right to continue her relationship with him if she wishes, but I do also think she needs to have all the information to hand before she does that. I've written her a message, I'd have to send it via Facebook so it's possible she won't even see it. It's not nasty at all, I don't want to hurt her. I've offered to give her copies of messages if she wishes to see them, I've given her my email if she wants to continue the conversation. I haven't sent the message yet.
Do I send the message?