CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Right adjoints preserve monomorphisms and left adjoints preserve epimorphisms

Statement

Every right adjoint preserves monomorphisms, and every left adjoint preserves epimorphisms.

Facts & Assumptions

Given: A morphism $f : x \to y$ and an adjunction whose right or left adjoint is applied to it.

[F1]

A morphism $f$ is monic when $f g = f h$ implies $g = h$ for every parallel pair, and epic when $g f = h f$ implies $g = h$ for every parallel pair (Monomorphism and epimorphism by left and right cancellation).

[L1]

Right adjoints preserve every existing limit (Right adjoints preserve every limit that exists).

[L2]

Left adjoints preserve every existing colimit (Left adjoints preserve every colimit that exists).

Proof

technique · direct

1.1

If $f$ is monic, the commutative square with vertex $x$ , both maps into the two copies of $x$ equal to $1_{x}$ , and both maps from those copies to $y$ equal to $f$ , is a pullback: a pair $g, h : z \to x$ with $f g = f h$ has the unique mediating map $g = h$ .

1.2

Conversely, if that self-square is a pullback and $f g = f h$ , then both $g$ and $h$ are mediating maps for the same cone, so pullback uniqueness gives $g = h$ ; hence $f$ is monic.

2.1

A right adjoint preserves the pullback in step 1.1 by [L1], and step 1.2 then says that the image of $f$ is monic.

step 1.1step 1.2L1

3.1

Passing to opposite categories turns epimorphisms into monomorphisms and a left adjoint into a right adjoint; equivalently, apply [L2] to the dual pushout characterization. Thus left adjoints preserve epimorphisms.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 9 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

Emily Riehl, Category Theory in Context, 2nd ed., Exercise 4.6.vi (standard reference, not scraped)