Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to show that if $C$ is a category and $A\in\operatorname{object} C$ is an initial object and $f \in \operatorname{Mor}(D,A)$ for $D \in \operatorname{object} C$, then $f$ is a retraction morphism in $C$.

By definition of initial objects $\lvert\operatorname{Mor} (A,C)\rvert=1 $ for all $A \in \operatorname{object} C$, then for $D\in \operatorname{object} C$ $\exists! g \in \operatorname{Mor}(A,D)$ s.t. $f\circ g\colon A\longrightarrow A $. How can i show that $f\circ g= I_{A}$?

share|cite|improve this question
up vote 2 down vote accepted

Take $D=A$ in the claim about the existence of a unique arrow $A\to D$. Thus there is a unique arrow $A\to A$. Clearly, $I_A$ is that unique arrow. So, $f\circ g=I_A$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.