Consider the sets:
Y = {y}
Z = {z}.
This is their function space: the space of functions from Y to Z:
ZY = {{y→z}}.
We can make it into an exponential object by equipping it with a morphism:
eval:(ZY × 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→ZY 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→ZY = {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:
You can read more about exponential objects at Wikipedia's Exponential object page. I have used the same notation.