Everything Points Here

At a glance

• Part 1a: There are 0, 1, and 8 functions from 𝐴 to ∅, 𝟏, and 𝐵, respectively.
• Parts 1b–1c: Every singleton is terminal in 𝐒𝐞𝐭; the empty set and a two-element set are not.
• Part 2: The unique arrows 𝑓 :𝑇 →𝑈 and 𝑔 :𝑈 →𝑇 satisfy
  𝑔∘𝑓=id𝑇,𝑓∘𝑔=id𝑈⁡.
  Moreover, 𝑓 is the only isomorphism 𝑇 →𝑈.

Key idea: uniqueness forces the inverse equations

Terminality is a statement about exactly one arrow into an object. If a terminal object is also used as the source, there is exactly one arrow from that object to itself. Its identity arrow is one such arrow, so every other arrow with the same source and target must equal the identity.

1. Test the definition in sets

Let

1a. Count the functions

There is no function 𝐴 →∅. A function must assign an output to each of 𝑟,𝑠,𝑡, but the empty target contains no possible output. The count is 0.

There is exactly one function 𝐴 →𝟏. Every input is forced to go to the only target element:

For a function 𝐴 →𝐵, each of the three inputs independently chooses either 0 or 1. Therefore the count is

1b. Prove that a singleton is terminal

Let 𝑋 be any set. A function 𝑋 →𝟏 has only one possible rule:

This rule defines a function, and no different function is possible because the target contains no other value. When 𝑋 =∅, the empty function is still the unique function to 𝟏. Thus every set has exactly one function to 𝟏, so 𝟏 is terminal in 𝐒𝐞𝐭.

1c. Rule out the other targets

There is no function {𝑟} →∅, so ∅ fails the existence part of terminality.

There are two functions {𝑟} →𝐵: one sends 𝑟 to 0, and the other sends 𝑟 to 1. Thus 𝐵 fails the uniqueness part of terminality.

2. Prove the theorem

Suppose 𝑇 and 𝑈 are terminal.

Since 𝑈 is terminal, there is exactly one arrow

Since 𝑇 is terminal, there is exactly one arrow

2b. The arrows are inverses

The composite 𝑔 ∘𝑓 and the identity id𝑇 are both arrows 𝑇 →𝑇. Terminality of 𝑇, applied with source 𝑇, says there is exactly one such arrow. Therefore

Similarly, 𝑓 ∘𝑔 and id𝑈 are both arrows 𝑈 →𝑈. Terminality of 𝑈 forces

Thus 𝑓 is an isomorphism with inverse 𝑔.

2c. The isomorphism is unique

Let ℎ :𝑇 →𝑈 be any isomorphism. It is, in particular, an arrow from 𝑇 into terminal object 𝑈. There is exactly one such arrow, namely 𝑓. Hence

This proves existence and exclusion: 𝑇 and 𝑈 are connected by exactly one isomorphism. Terminal objects are therefore unique up to unique isomorphism.

Check against the definitions

The construction uses only arrows guaranteed by terminality. Both required inverse equations were proved, and any competing isomorphism was ruled out by the same uniqueness condition. Notice that different singleton sets in 𝐒𝐞𝐭 need not be literally equal; the theorem correctly says they are uniquely isomorphic.

Another facet

Category theory often identifies an object by the arrows it must receive or send. When such a description forces exactly one comparison isomorphism, the resulting object is canonical in structure even when it is not literally equal to another representative.

Teaching notes

• A learner who draws arrows out of a terminal object toward every object has reversed the definition. Terminal arrows point into the terminal object.
• The phrase “unique up to unique isomorphism” is stronger and more precise than saying terminal objects are equal.
• The decisive move is to apply terminality with the terminal object itself as the source.