TITLE: About contractibility of certain categories
QUESTION [13 upvotes]: Let $\mathcal{C}$ be an ordinary 1-category and suppose that there exists some object $X \in \mathcal{C}$ such that the following conditions are satisfied,
(1) For every $C \in \mathcal{C}$ we have $\operatorname{Hom}(X,C)\neq \emptyset$ , $\operatorname{Hom}(C,X)\neq \emptyset$.
(2) $\operatorname{Hom}(X,X)=*$.
It is easy to see that its 1-groupoidification must be categorically equivalent to the point. 
I would like to know if these 2 conditions already imply contractibility of the nerve of $\mathcal{C}$.

REPLY [9 votes]: A counterexample is Connes’ cyclic category $\Lambda$: objects are $\langle n\rangle$ for all $n\in\mathbb N$; arrows $\langle n\rangle \to \langle m\rangle$ are homotopy classes of monotone, degree-$1$ maps $(S^1,\mu_{n+1})\to (S^1, \mu_{m+1}$) between circles with marked roots of unity.
Its classifying space is known to be $\mathrm{B} S^1$ but it has an object $\langle 0\rangle$ satisfying your conditions because $\Lambda(\langle n\rangle, \langle 0\rangle ) \cong \Lambda (\langle 0\rangle, \langle n\rangle ) \cong \{0,\dots , n\}$ for all $n\in\mathbb N$