{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ and $G \\colon \\mathcal{D} \\to \\mathcal{C}$ be functors.\n\n1. Let $\\varphi$ be an [[adjunction|TZHF6ZM92S]] between $F$ and $G$, with [[unit and counit|MYGWT305AB]] $\\eta$ and $\\varepsilon$.\nThen $\\eta$ and $\\varepsilon$ are [[natural transformations|KYCPCC57FT]] and satisfy $\\varepsilon_{F c} \\circ F(\\eta_c) = \\id_{F c}$ and $G(\\varepsilon_d) \\circ \\eta_{G d} = \\id_{G d}$.\n2. Conversely, let $\\eta \\colon \\mathrm{Id}_{\\mathcal{C}} \\Rightarrow G F$ and $\\varepsilon \\colon F G \\Rightarrow \\mathrm{Id}_{\\mathcal{D}}$ be natural transformations that satisfy these two triangle identities.\nThen $\\varphi(g) = G(g) \\circ \\eta_c$, for $g \\colon F c \\to d$, is an adjunction.\nIts inverse is $f \\mapsto \\varepsilon_d \\circ F(f)$ for $f \\colon c \\to G d$.\n3. These two constructions are inverse to each other.\n\n## Remarks\n\nFor part 1, naturality of $\\varphi$ gives $\\varphi(g) = G(g) \\circ \\eta_c$ and $\\varphi^{-1}(f) = \\varepsilon_d \\circ F(f)$.\nApplying these to identities gives the triangle identities.\nFor part 2, the triangle identities are exactly what is needed for the two maps to be inverse to each other, and naturality is a direct check.\n\nExample: for sets, $F = - \\times Y$ and $G = (-)^Y$, the unit sends $x \\in X$ to the map $y \\mapsto (x, y)$.\nThe counit is evaluation $Z^Y \\times Y \\to Z$.\nThe triangle identities then say that currying and evaluating undo each other.\n\n## Explanations\n\n### L1 human\n\nThere are two ways to describe an adjunction.\nOne matches maps $F c \\to d$ with maps $c \\to G d$.\nThe other gives two families of maps, the unit and the counit, which satisfy two simple equations.\nThe theorem says that the two descriptions carry the same information.\n\n### L2 human\n\nThe hom-set form is useful for recognizing adjunctions, since it is often a familiar bijection such as currying or the universal property of free objects.\nThe unit and counit form is better for computation, because it is a finite set of equations between natural transformations.\nIt also makes sense in any 2-category, so it is the form used to define adjunctions in general.\nA third equivalent form says that each $\\eta_c$ is an initial object of the comma category $c \\downarrow G$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "TZHF6ZM92S",
      "MYGWT305AB"
    ],
    "kind": "statement",
    "msc": [
      "18A40"
    ],
    "name": "Hom-set characterization of adjunctions",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Adjoint_functors"
      }
    ],
    "tag": "4KCECXYW81",
    "uses": [
      "TZHF6ZM92S",
      "MYGWT305AB",
      "KYCPCC57FT"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/4KCECXYW81/",
  "links": [
    {
      "field": "about",
      "target": "TZHF6ZM92S"
    },
    {
      "field": "about",
      "target": "MYGWT305AB"
    },
    {
      "field": "uses",
      "target": "TZHF6ZM92S"
    },
    {
      "field": "uses",
      "target": "MYGWT305AB"
    },
    {
      "field": "uses",
      "target": "KYCPCC57FT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4KCECXYW81.md",
  "msc": [
    "18A40"
  ],
  "name": "Hom-set characterization of adjunctions",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4KC/4KCECXYW81.md",
  "status": null,
  "tag": "4KCECXYW81",
  "uri": "m2://4KCECXYW81",
  "url": "https://tamnd.github.io/math2/wiki/hom-set-characterization-of-adjunctions/",
  "writing": {
    "tag": "4KCECXYW81",
    "kind": "statement",
    "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
      }
    ]
  }
}