About contractibility of certain categories 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 $\mathcal{C}$.
 A: Let $\mathcal{C}$ be the category with two objects $X$ and $Y$, generated by morphisms $\alpha_1,\alpha_2:X\to Y$ and $\beta_1,\beta_2:Y\to X$ subject to relations $\beta_i\alpha_j=\text{id}_X$ for all $i,j$.
So the only non-identity morphisms other than $\alpha_1,\alpha_2,\beta_1,\beta_2$ are
the compositions $\alpha_i\beta_j:Y\to Y$.
According to my calculations, the nerve is homotopy equivalent to a $2$-sphere, but here is a proof that at least it has the cohomology with coefficients in a field $k$ of a $2$-sphere, and is therefore not contractible.
I think it is a standard fact that the cohomology $H^n(B\mathcal{C},k)$ of the classifying space of $\mathcal{C}$ is equal to the extension group $\text{Ext}^n(\mathbf{k},\mathbf{k})$ in the category of functors from $\mathcal{C}$ to $k$-vector spaces, where $\mathbf{k}$ is the constant functor taking the value $k$.
For each object $V$ of $\mathcal{C}$, there is a projective functor $P_V$ whose value on an object $U$ is the vector space with basis $\mathcal{C}(V,U)$, and a morphism $\alpha:V\to V'$ induces a morphism of functors $\alpha^\ast:P_{V'}\to P_V$ by composition.
A straightforward calculation shows that
$$0\longrightarrow P_X\oplus P_X\stackrel{\pmatrix{\beta^\ast_1\\\beta^\ast_2}}{\longrightarrow} P_Y\stackrel{\alpha^\ast_1-\alpha^\ast_2}{\longrightarrow} P_X\longrightarrow\mathbf{k}\longrightarrow0$$
is a projective resolution of the constant functor, and applying the functor $\text{Hom}(-\mathbf{k})$ to the projective terms to calculate $\text{Ext}^*(\mathbf{k},\mathbf{k})$ gives
$$k\stackrel{0}{\longrightarrow}k\stackrel{\pmatrix{1&1}}{\longrightarrow}k^2\longrightarrow0,$$
so $\text{Ext}^*(\mathbf{k},\mathbf{k})$ is one-dimensional in degrees zero and two, and zero in all other degrees.
A: 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 $n\in\mathbb N$
