The Canonical Tag Checker loads a page and reads its canonical tag. It runs 5 checks: a canonical is present, there is only one, it points at the page itself, it uses an absolute URL, and the page is not also marked noindex. You also get the canonical URL, the number of canonical tags found, and the final URL after any redirects.
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The headline says one of three things: the canonical points at itself, it points elsewhere, or there is no canonical tag. The checks underneath show the detail.
The tool reads the canonical tag in the page HTML. A canonical sent in an HTTP header is not checked.
Canonical present warns when there is no tag, which leaves search engines to pick the preferred URL themselves. Exactly one canonical fails when the page has 2 or more tags, since conflicting canonicals may be ignored.
Self-referencing passes when the canonical matches the final URL of the page, ignoring a trailing slash. It warns when the canonical points at a different URL. That is correct on a duplicate page you want consolidated, and a problem on a page you want indexed.
Absolute URL fails when the canonical is a relative path. Not also noindex fails when the page carries both a canonical and a noindex robots meta tag, since the two send opposite signals.
Peak Answer runs checks like these every week, then writes the content and gets the placements.