{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "4KCECXYW81"
    },
    {
      "field": "uses",
      "target": "4KCECXYW81"
    },
    {
      "field": "uses",
      "target": "DF5N0PK3ZS"
    },
    {
      "field": "defines",
      "target": "JY9K3YXW0X"
    }
  ],
  "body": "\n## Summary\n\nThe unit and the counit of an adjunction $F \\dashv G$ are the two [[natural transformations|KYCPCC57FT]] $\\eta \\colon \\mathrm{Id} \\Rightarrow G F$ and $\\varepsilon \\colon F G \\Rightarrow \\mathrm{Id}$ that it determines.\nThey satisfy the two triangle identities, and conversely they determine the [[adjunction|TZHF6ZM92S]].\nThe unit and the counit give the [[monad|FF6XXA6D1A]] and the comonad of the adjunction.\n\n## Explanations\n\n### L1 human\n\nFor the free group and the forgetful functor, the unit sends each element of a set $X$ to the corresponding generator of the free group on $X$.\nThe counit takes the free group on the elements of a group $H$ and evaluates each word by multiplying in $H$.\nSo the unit says how to put things in, and the counit says how to compute the result.\nThe triangle identities say that putting in and then computing gives back what you started with.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A40"
    ],
    "name": "Unit and counit",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Adjoint_functors"
      }
    ],
    "tag": "MYGWT305AB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/MYGWT305AB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/MYGWT305AB.md",
  "msc": [
    "18A40"
  ],
  "name": "Unit and counit",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MYG/MYGWT305AB.md",
  "status": null,
  "tag": "MYGWT305AB",
  "uri": "m2://MYGWT305AB",
  "url": "https://tamnd.github.io/math2/wiki/unit-and-counit/",
  "writing": {
    "tag": "MYGWT305AB",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}