Consider the sets:

Y = {y}

Z = {z}.

This is their function space: the space of functions from Y to Z:

Z^{Y}= {{y→z}}.

We can make it into an exponential object by equipping it with a morphism:

eval:(Z^{Y}× Y)→Z = {⟨{y→z},y⟩→z}

Then the following applies
for any object X and
morphism g:(X × Y)→Z.

There is a
unique morphism λg:X→Z^{Y} such that
the following diagram commutes:

[diagram in VRML]

Let me demonstrate. Because the above
statement works for *any* X and g, I shall
demonstrate by generating an X and g at random:

X = {x}

g:(X × Y)→Z = {⟨x,y⟩→z}

And watch! λg is this:

λg:X→Z^{Y}= {x→{y→z}}.

This is a listing of the objects and arrows involved:

X = {x}

X × Y = {⟨x,y⟩}

Y = {y}

Z = {z}

Z^Y = {{y→z}}

Z^Y × Y = {⟨{y→z},y⟩}

λg:X→Z^Y = {x→{y→z}}

λg × idY:X × Y→Z^Y × Y = {⟨x,y⟩→⟨{y→z},y⟩}

( λg × idY ) ; eval:X × Y→Z = {⟨x,y⟩→z}

eval:Z^Y × Y→Z = {⟨{y→z},y⟩→z}

g:X × Y→Z = {⟨x,y⟩→z}

idY:Y→Y = {y→y}

And finally, this is a step-by-step guide to how I constructed these objects and arrows:

- Generate a random set Y.
- Generate a random set Z.
- Calculate the set Z
^{Y}, the set of all functions from Y to Z. - Generate a random set X.
- Calculate X × Y, the Cartesian product of X and Y.
- Generate a random map g from X × Y to Z.
- Calculate λg, by currying g.
- Calculate id
_{Y}, the identity map from Y to Y. - Calculate λg × id
_{Y}, the product map of λg and id_{Y}. - Calculate Z
^{Y}× Y, the Cartesian product of Z^{Y}and Y. - Calculate eval, the
evaluation map from Z
^{Y}× Y. For each pair ⟨map_from_Y_to_Z,y.⟩, where map_from_Y_to_Z ∈ Z^{Y}and y ∈ Y, eval maps that pair to map_from_Y_to_Z(y).

You can read more about exponential objects at Wikipedia's Exponential object page. I have used the same notation.