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

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Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$

Statement

For every set $A$ , the product functor $- \times A : S e t \to S e t$ is left adjoint to the function-set functor $(-)^{A}$ . Naturally in $X$ and $Y$ ,

$S e t (X \times A, Y) ≅ S e t (X, Y^{A}) .$

The bijection sends $h$ to $\hat{h} (x) (a) = h (x, a)$ and sends $k$ to $\overset{ˇ}{k} (x, a) = k (x) (a)$ .

Facts & Assumptions

Given: Sets $A, X, Y$ .

[F1]

The functions $A \to Y$ form the set $Y^{A}$ (The set $B^{A}$ of all functions $A \to B$ ).

[F2]

The function collection between two sets is a set (For sets $A$ and $B$ the collection of all functions $A \to B$ is a set, being a subset of $P (A \times B)$ ).

[F3]

The cartesian product $X \times A$ consists of ordered pairs $(x, a)$ with $x \in X$ , $a \in A$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ).

[F4]

Sets and functions form the locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[L1]

Natural hom-set bijections determine an adjunction (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1

For $h : X \times A \to Y$ , define $\hat{h} : X \to Y^{A}$ by $\hat{h} (x) (a) = h (x, a)$ ; [F1] and [F3] make this a well-defined function.

F1F3construct

1.2

For $k : X \to Y^{A}$ , define $\overset{ˇ}{k} : X \times A \to Y$ by $\overset{ˇ}{k} (x, a) = k (x) (a)$ .

F1F3construct

2.1

For all $(x, a)$ , $\overset{ˇ}{\hat{h}} (x, a) = h (x, a)$ , so function extensionality gives $\overset{ˇ}{\hat{h}} = h$ ; similarly $\hat{\overset{ˇ}{k}} (x) (a) = k (x) (a)$ gives $\hat{\overset{ˇ}{k}} = k$ .

step 1.1step 1.2F2

2.2

Precomposition in $X$ and postcomposition in $Y$ commute with evaluation at $(x, a)$ , so the bijection is natural in both variables.

step 1.1step 1.2algebra

3.1

Since $S e t$ is locally small by [F4], [L1] applies and gives $- \times A ⊣ (-)^{A}$ . The formulas also cover $A = \emptyset$ without exception.

step 2.1step 2.2F4L1∎

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: 35 results over 14 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., Example 4.4.9 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Example 2.1.6 (standard reference, not scraped)

Currying gives the adjunction −×A⊣(−)A−×A⊣(−)A in SetSet