You can use Amazon Cognito User Pools federation by adding a sign-in through a SAML IdP (among others). Just to note that this is different from Amazon Cognito Identity Pools (Federated Identities) flow. Building ADFS Federation for your Web App using Amazon Cognito User Pools blog post provides end-to-end walk through.
As for SAML IdP identifier to automatically redirect the user to relevant IdP and UI customization, see Choosing SAML Identity Provider Names in the documentation.
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