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# Exercise 30 - Chapter 3

edited June 2018

Show that in any category $$C$$, for any object $$c \in C$$, the identity $$id_c$$ is an isomorphism.

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1.

Per the definition of isomorphism, it is sufficient to find a morphism $$g\in\mathcal{C}(c,c)$$) such that $$g\circ\text{id}_c=\text{id}_c$$ and $$\text{id}_c\circ g=\text{id}_c$$. For any object in any category, the identity morphism is guaranteed to exist and $$\text{id}_c\circ\text{id}_c=\text{id}_c$$. So, $$\text{id}_c$$ is an isomorphism with inverse given by $$\text{id}_c$$.

Comment Source:Per the definition of isomorphism, it is sufficient to find a morphism \$$g\in\mathcal{C}(c,c)\$$) such that \$$g\circ\text{id}_c=\text{id}_c\$$ and \$$\text{id}_c\circ g=\text{id}_c\$$. For any object in any category, the identity morphism is guaranteed to exist and \$$\text{id}_c\circ\text{id}_c=\text{id}_c\$$. So, \$$\text{id}_c\$$ is an isomorphism with inverse given by \$$\text{id}_c\$$.