Ways to set up two-way authentication based on client certificates with a third party service?

Question

Digital certificates

Can you help in the direction of finding information and what to read ...
Bottom line:
There is a system (A) that provides some kind of API, but it is with two-way authentication based on client certificates.
And my service is in the role of an intermediary. provides some UI.
That is, the client (user) has everything necessary for working with certificates on his machine.
When generating requests, through through my site, forms and sends a request to the service (A).

How can I set up two-way authentication based on client certificates. with service (A), through my website?

Answer 1

This is not the first time you have asked and I still can’t figure it out - what is the meaning of your site for the user? Certificates are needed in order to ensure that no one interferes with a two-way connection, no third person listens to it. And at you MitM such turns out, yes? Or is the connection established from your site?