A terminal object is isomorphic to an initial object if there is a morphism from T to I

up:: 028 MOC Category Theory

We have that every Initial Object $I$ has a unique morphism coming out of it towards any other object in a Category $C$, and that every Terminal Object $T$ has a unique object coming into it. Thus, there is only one morphism from $I$ to $T$; call it $I\stackrel{f}{\to }T$.

In the case where $\exists \phi \in Hom(T,I)$¹, we can see that

$$\{\begin{array}{ll}{1}_I& =f;\phi \\ 1_T& =\phi ;f\end{array}$$

Therefore, we have that $\phi$ is the inverse of $f$ ─ therefore, $f$ is an isomorphism.

Corollaries

When there is a morphism from a terminal object to an initial object, we have that they are isomorphic. In this case, they will be isomorphic to a Zero Object. Thus, there can only be one zero object (up to isomorphism) in a category.

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References

• KASHIWARA, Masaki; SCHAPIRA, Pierre, Categories and Sheaves, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

Footnotes

1. Note that this doesn’t make $T$ stop being a terminal object; it is an additional property that it possesses, aside from being terminal. ↩