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How we can see that initial and terminal objects are unique upto unique isomorphism and where they exist?

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closed as off-topic by Martin Brandenburg, Brian Rushton, Daniel Rust, Sami Ben Romdhane, azimut Jan 5 at 19:55

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The first part of your question is very easy, please try it yourself. The second part is not specific. –  Martin Brandenburg Jan 5 at 18:45
    
There is more or less only one thing you can do with the knowledge that something is an initial object. Do it. –  Hurkyl Jan 5 at 18:46

2 Answers 2

up vote 2 down vote accepted

Hint: Let $A$ and $A'$ be initial objects in your category. Let $f$ and $g$ be the unique morphisms between them according to the following diagram $$ A\stackrel{f}{\to}A'\stackrel{g}{\to}A\stackrel{f}{\to}A' $$ So $gf$ is a morphism from $A$ to itself, and $fg$ is a morphism from from $A'$ to itself. Since $A$ and $A'$ are initial objects, what must $fg$ and $gf$ be? What does this tell you about $f$ and $g$?

Since $A$ is initial, the only morphism from $A$ to itself is $1_A$, and similarly for $A'$. So $gf=1_A$, and $fg=1_{A'}$ which implies $f$ and $g$ are isomorphisms.

The terminal case is dual and I leave it to you.

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Use the definition of initial/terminal object with itself.

Somehow, this always reminds me of a proof that isosceles triangles ABC with $AB=BC$ have angles A and C equal. The proof that is applying SAS triangle equality criterion with the triangle ABC and CBA.

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