You need to set up a Claims Provider trust on ADFS pointing to your Identity Provider. On your Identity Provider, set up an RP trust back to ADFS.
Also the ASP.NET web application needs to use ADFS as it's STS.
And the claims need to be set as pass through.
Refer : ADFS 2.0 By Example - Part1: ADFS as IP-STS and R-STS
Update:
On the IP side, that's their problem. On your side, configure the trust manually.
Refer: ADFS : SAML configuration parameters.
These questions relate directly to the input parameters.
You need to export the ADFS token signing key (no need to export the private key) and send that to them. They need to send you their certificate. Once you have set up the trust manually, click on the certificate tab and import their certificate.