docs: update middleware.ts references to proxy.ts for Next.js 16

[F-47]

Summary

Next.js 16 renamed middleware.ts to proxy.ts and the exported function from middleware to proxy. The proxy.ts file also now runs on the Node.js runtime instead of the edge runtime.

This PR updates all documentation references across 9 files to reflect the new convention:

• installation.mdx - Updated setup step to use proxy.ts with export { auth as proxy }
• index.mdx - Updated homepage code example
• protecting.mdx - Renamed "Next.js Middleware" section to "Next.js Proxy", updated all code examples
• edge-compatibility.mdx - Updated split config examples, added callout noting proxy.ts runs on Node.js (edge workarounds may no longer be needed)
• migrating-to-v5.mdx - Updated migration examples and edge compatibility section
• role-based-access-control.mdx - Updated callout and resource links
• passkey.mdx - Updated edge compatibility note
• prisma.mdx - Updated middleware references in compatibility warnings
• concepts/index.mdx - Updated FAQ about middleware/proxy compatibility

All pages include backward-compatibility notes for users on older Next.js versions who still use middleware.ts.

References

• Next.js 16 Blog Post
• Next.js proxy.ts API Reference
• Middleware to Proxy Migration Guide

Test plan

• Verify all internal doc links (e.g. #nextjs-proxy) resolve correctly
• Verify code examples are syntactically correct
• Verify callouts render properly in the docs site
• Check that backward-compatibility notes are clear for users on older Next.js versions

[vercel]

The latest updates on your projects. Learn more about Vercel for GitHub.

Project	Deployment	Actions	Updated (UTC)
auth-docs	Ready	Preview, Comment	Feb 3, 2026 9:13am

1 Skipped Deployment

Project	Deployment	Actions	Updated (UTC)
next-auth-docs	Ignored	Preview	Feb 3, 2026 9:13am

[vercel]

@FGameIn is attempting to deploy a commit to the authjs Team on Vercel.

A member of the Team first needs to authorize it.