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LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A morphism is an isomorphism exactly when postcomposition, equivalently precomposition, induces bijections on every hom-collection

Statement

For a morphism f:A→Bf:A\to B, the following are equivalent: ff is an isomorphism; for every object XX, postcomposition f∘−:C(X,A)→C(X,B)f\circ-:\mathcal C(X,A)\to\mathcal C(X,B) is bijective; and for every XX, precomposition −∘f:C(B,X)→C(A,X)-\circ f:\mathcal C(B,X)\to\mathcal C(A,X) is bijective.

Facts & Assumptions

Given: A morphism f:A→Bf:A\to B in a category C\mathcal C.

[L1]

Isomorphisms have two-sided inverses (Isomorphism, groupoid, and connected category), and a map is bijective exactly when it has a two-sided inverse (Injection, surjection, bijection).

[L2]

Reversing arrows exchanges postcomposition with precomposition (Every theorem about categories has a formal dual obtained by reversing morphisms and composition).

Proof

technique · direct
1.1

If ff has inverse f−1f^{-1}, postcomposition by f−1f^{-1} is a two-sided inverse to postcomposition by ff, and similarly precomposition by f−1f^{-1} inverts precomposition by ff; both maps are bijections.

givenL1
2.1

Conversely, suppose every postcomposition map is bijective. Surjectivity at X=BX=B gives g:B→Ag:B\to A with f∘g=1Bf\circ g=1_B; injectivity at X=AX=A applied to f∘(g∘f)=f=f∘1Af\circ(g\circ f)=f=f\circ1_A gives g∘f=1Ag\circ f=1_A, so ff is an isomorphism.

step 1.1L1
3.1

The identical argument in Cop\mathcal C^{\mathrm{op}}, using [L2], proves that bijectivity of every precomposition map also characterises isomorphisms.

step 2.1L2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources