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# Question 1.3 - Is it an isomorphism?

edited January 19

Is it an isomorphism?

Suppose that someone tells you that their category $$C$$ has two objects $$c,d$$ and two non-identity morphisms, $$f : c \to d$$ and $$g : d \to c$$, but no other morphisms. Does $$f$$ have to be the inverse of $$g$$, i.e. is it forced by the category axioms that $$g◦f = id_c$$ and $$f◦g= id_d$$?

(i) $$f◦g$$ is a morphism from $$d \to d$$. Since the only morphism in $$C$$ from $$d \to d$$ is $$id_d$$ we must have $$f◦g= id_d$$.
(ii) $$g◦f$$ is a morphism from $$c \to c$$. Since the only morphism in $$C$$ from $$c \to c$$ is $$id_c$$ we must have $$g◦f= id_c$$.
Comment Source:My argument is as follows : (i) \$$f◦g\$$ is a morphism from \$$d \to d \$$. Since the only morphism in \$$C \$$ from \$$d \to d \$$ is \$$id_d \$$ we must have \$$f◦g= id_d \$$. (ii) \$$g◦f\$$ is a morphism from \$$c \to c \$$. Since the only morphism in \$$C \$$ from \$$c \to c \$$ is \$$id_c \$$ we must have \$$g◦f= id_c \$$.